Baryzentrische Koordinaten

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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Dieser_Artikel"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" id="bksicon" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Dieser Artikel beschreibt baryzentrische Koordinaten in der <a href="/wiki/Geometrie" title="Geometrie">Geometrie</a>. In der <a href="/wiki/Astronomie" title="Astronomie">Astronomie</a> bezeichnet man Koordinaten als baryzentrisch, die sich auf ein <a href="/wiki/Schwerpunktsystem" title="Schwerpunktsystem">Schwerpunktsystem</a> eines Systems von <a href="/wiki/Himmelsk%C3%B6rper" class="mw-redirect" title="Himmelskörper">Himmelskörpern</a> beziehen, vgl. auch <a href="/wiki/Astronomische_Koordinatensysteme" title="Astronomische Koordinatensysteme">Astronomische Koordinatensysteme</a>.</div> </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-def-2d.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Baryzentr-ko-def-2d.svg/280px-Baryzentr-ko-def-2d.svg.png" decoding="async" width="280" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Baryzentr-ko-def-2d.svg/420px-Baryzentr-ko-def-2d.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Baryzentr-ko-def-2d.svg/560px-Baryzentr-ko-def-2d.svg.png 2x" data-file-width="261" data-file-height="243" /></a><figcaption>Die baryzentrischen Koordinaten eines Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> (blau) sind die Verhältnisse dreier Massen in den Ecken eines Dreiecks (rot), deren <a href="/wiki/Massenmittelpunkt" title="Massenmittelpunkt">Schwerpunkt</a> (Massenmittelpunkt) der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> ist. In diesem Beispiel hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2:4:5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>:</mo> <mn>4</mn> <mo>:</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2:4:5)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc89e633bf748b1aeb30c196834048e164fbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle (2:4:5)}">.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-def-1d-z.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Baryzentr-ko-def-1d-z.svg/260px-Baryzentr-ko-def-1d-z.svg.png" decoding="async" width="260" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Baryzentr-ko-def-1d-z.svg/390px-Baryzentr-ko-def-1d-z.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Baryzentr-ko-def-1d-z.svg/520px-Baryzentr-ko-def-1d-z.svg.png 2x" data-file-width="261" data-file-height="176" /></a><figcaption>Die Verbindung zwischen Physik und Geometrie liefert die Gleichung des <a href="/wiki/Hebelgesetz" class="mw-redirect" title="Hebelgesetz">Hebelgesetzes</a>: Danach ist das Verhältnis der Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd2f8098df7628f42d6b2f8fdb7450122aefec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.223ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2}}"> gleich dem Verhältnis der Strecken <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{2},l_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{2},l_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9831616bb6ed3446bb72c82013921800d85471a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.529ex; height:2.509ex;" alt="{\displaystyle l_{2},l_{1}}">, die die Lage des Schwerpunktes beschreiben.</figcaption></figure> Baryzentrische Koordinaten (auch homogene baryzentrische Koordinaten) dienen in der <a href="/wiki/Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> und in der <a href="/wiki/Geometrie" title="Geometrie">Geometrie</a> dazu, die Lage von <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkten</a> in Bezug auf eine gegebene <a href="/wiki/Strecke_(Geometrie)" title="Strecke (Geometrie)">Strecke</a>, ein gegebenes <a href="/wiki/Dreieck" title="Dreieck">Dreieck</a>, ein gegebenes <a href="/wiki/Tetraeder" title="Tetraeder">Tetraeder</a> oder allgemeiner ein gegebenes <a href="/wiki/Simplex_(Mathematik)" title="Simplex (Mathematik)">Simplex</a> zu beschreiben. Ebene baryzentrische Koordinaten eines Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> kann man sich als Verhältnisse von drei Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},m_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},m_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3964c8bc26f931bf70ee2d8bd12a266f2e817c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.352ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},m_{3}}"> vorstellen, die sich in den Ecken eines vorgegebenen Dreiecks befinden und deren <a href="/wiki/Massenmittelpunkt" title="Massenmittelpunkt">Schwerpunkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> ist (siehe Bild). Da es dabei nur auf Verhältnisse ankommt, schreibt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{1}:m_{2}:m_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{1}:m_{2}:m_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33805be288a83ef5b5c4acac3aa4a8ab429fdca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.968ex; height:2.843ex;" alt="{\displaystyle (m_{1}:m_{2}:m_{3})}">. Sind alle Massen gleich, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> der <a href="/wiki/Geometrischer_Schwerpunkt" title="Geometrischer Schwerpunkt">geometrische Schwerpunkt</a> des Dreiecks und hat die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8503c34e46b515c671b4ae950d73dcc3d9722f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle (1:1:1)}">. Ihre geometrische Bedeutung erhalten die baryzentrischen Koordinaten durch die folgenden Eigenschaften: Im 1-Dimensionalen ist das Massenverhältnis gleich einem Verhältnis von Teilstrecken (siehe 2. Bild), im 2-Dimensionalen sind die Massenverhältnisse gleich Flächenverhältnissen von Teildreiecken. Baryzentrische Koordinaten wurden zuerst von <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">A. F. Möbius</a> 1827 in seinem Buch Der baryzentrische Calcul eingeführt.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> Sie sind ein Spezialfall <a href="/wiki/Homogene_Koordinaten" title="Homogene Koordinaten">homogener Koordinaten</a>. Ein wesentlicher Unterschied zu den üblichen homogenen Koordinaten, z. B. in der Ebene, ist die Beschreibung der Ferngerade durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+x_{2}+x_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}+x_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a37e9c6b819f4da6042404adb9a11094fcf97a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.093ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}+x_{3}=0}"> statt durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55547972561662fda8d82465866be57f729271c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{3}=0}">. Insbesondere in der <a href="/wiki/Dreiecksgeometrie" title="Dreiecksgeometrie">Dreiecksgeometrie</a> spielen die baryzentrischen Koordinaten, neben den <a href="/wiki/Trilineare_Koordinaten" title="Trilineare Koordinaten">trilinearen Koordinaten</a>, eine wesentliche Rolle. Überall, wo es um Verhältnisse von Strecken geht, wie zum Beispiel in dem <a href="/wiki/Satz_von_Ceva" title="Satz von Ceva">Satz von Ceva</a>, sind sie ein geeignetes Werkzeug. Aber nicht nur in der Geometrie, sondern auch im Bereich des <a href="/wiki/CAD" title="CAD">computer-aided Design</a> verwendet man sie zur Erzeugung von dreieckigen Flächenstücken, den dreieckigen <a href="/wiki/B%C3%A9zierfl%C3%A4che" class="mw-redirect" title="Bézierfläche">Bézierflächen</a>.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition_und_Eigenschaften">1 Definition und Eigenschaften</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Definition">1.1 Definition</a></li> <li class="toclevel-2 tocsection-3"><a href="#Eigenschaften">1.2 Eigenschaften</a> <ul> <li class="toclevel-3 tocsection-4"><a href="#Punkt_im_Simplex">1.2.1 Punkt im Simplex</a></li> <li class="toclevel-3 tocsection-5"><a href="#Massenmittelpunkt">1.2.2 Massenmittelpunkt</a></li> <li class="toclevel-3 tocsection-6"><a href="#Mittelpunkt_zweier_Punkte">1.2.3 Mittelpunkt zweier Punkte</a></li> <li class="toclevel-3 tocsection-7"><a href="#Existenz,_Eindeutigkeit_normierter_Koordinaten">1.2.4 Existenz, Eindeutigkeit normierter Koordinaten</a></li> <li class="toclevel-3 tocsection-8"><a href="#Unabhängigkeit_von_Nullpunkt_und_Skalierung">1.2.5 Unabhängigkeit von Nullpunkt und Skalierung</a></li> <li class="toclevel-3 tocsection-9"><a href="#Beispiel">1.2.6 Beispiel</a></li> <li class="toclevel-3 tocsection-10"><a href="#Vorteil,_Nachteil">1.2.7 Vorteil, Nachteil</a></li> <li class="toclevel-3 tocsection-11"><a href="#Unterschied_zu_anderen_homogenen_Koordinaten:_Beispiel_n=3">1.2.8 Unterschied zu anderen homogenen Koordinaten: Beispiel n=3</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Auf_einer_Gerade_(n=2,_Strecke)">2 Auf einer Gerade (n=2, Strecke)</a></li> <li class="toclevel-1 tocsection-13"><a href="#In_einer_Ebene_(n=3,_Dreieck)">3 In einer Ebene (n=3, Dreieck)</a> <ul> <li class="toclevel-2 tocsection-14"><a href="#Umrechnung_der_Koordinaten">3.1 Umrechnung der Koordinaten</a></li> <li class="toclevel-2 tocsection-15"><a href="#Geraden,_Schnittpunkte,_Parallelität">3.2 Geraden, Schnittpunkte, Parallelität</a></li> <li class="toclevel-2 tocsection-16"><a href="#Beziehung_zu_trilinearen_Koordinaten">3.3 Beziehung zu trilinearen Koordinaten</a></li> <li class="toclevel-2 tocsection-17"><a href="#Besondere_Punkte,_Eulergerade">3.4 Besondere Punkte, Eulergerade</a></li> <li class="toclevel-2 tocsection-18"><a href="#Satz_von_Ceva">3.5 Satz von Ceva</a></li> <li class="toclevel-2 tocsection-19"><a href="#Steiner-Ellipse,_Steiner-Inellipse">3.6 Steiner-Ellipse, Steiner-Inellipse</a></li> </ul> </li> <li class="toclevel-1 tocsection-20"><a href="#Im_Raum_(n=4,_Tetraeder)">4 Im Raum (n=4, Tetraeder)</a> <ul> <li class="toclevel-2 tocsection-21"><a href="#Berechnung_und_Eigenschaften">4.1 Berechnung und Eigenschaften</a></li> <li class="toclevel-2 tocsection-22"><a href="#Besondere_Punkte">4.2 Besondere Punkte</a></li> <li class="toclevel-2 tocsection-23"><a href="#Satz_von_Commandino">4.3 Satz von Commandino</a></li> <li class="toclevel-2 tocsection-24"><a href="#Hyperboloid_durch_die_Punkte_eines_Tetraeders">4.4 Hyperboloid durch die Punkte eines Tetraeders</a></li> </ul> </li> <li class="toclevel-1 tocsection-25"><a href="#Verallgemeinerte_baryzentrische_Koordinaten">5 Verallgemeinerte baryzentrische Koordinaten</a></li> <li class="toclevel-1 tocsection-26"><a href="#Baryzentrische_Interpolation">6 Baryzentrische Interpolation</a></li> <li class="toclevel-1 tocsection-27"><a href="#Literatur">7 Literatur</a></li> <li class="toclevel-1 tocsection-28"><a href="#Weblinks">8 Weblinks</a></li> <li class="toclevel-1 tocsection-29"><a href="#Einzelnachweise">9 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition_und_Eigenschaften">Definition und Eigenschaften</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition und Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition und Eigenschaften">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=2" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688990fd5b8f4c39c97e473225326f40987148ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.273ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}"> die Ortsvektoren der Ecken <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dotsc ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25602d4b1df2584ef4dd94474b2db32c4e970e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}}"> eines Simplex in einem <a href="/wiki/Affine_R%C3%A4ume" class="mw-redirect" title="Affine Räume">affinen Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}">. Der affine Raum hat dann die Dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}">. Falls es für einen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:\mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P:\mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4fa6445359d60373bd6597775147ffb576bfacf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.168ex; height:2.509ex;" alt="{\displaystyle P:\mathbf {p} }"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"> Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;a_{1},\dotsc ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;a_{1},\dotsc ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cb0eb13a716826c4511d31404eff897c781118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.556ex; height:2.009ex;" alt="{\displaystyle \;a_{1},\dotsc ,a_{n}}"> gibt, deren Summe nicht Null ist und die Gleichung <dl><dd>(G)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b9d68bde8e25d45f0acaf78395b1ec748cc463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.085ex; height:2.843ex;" alt="{\displaystyle \quad (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}\ ,}"></dd></dl> erfüllt, sagt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dotsc ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dotsc ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bda393afa1b6471d6a48ff74e876d1d02f4415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dotsc ,a_{n}}"> sind baryzentrische Koordinaten des Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> bezüglich der Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dotsc X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd19b356ab91b8e60aacf746e049938b47b498c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.266ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc X_{n}}"> und schreibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(a_{1}:\dotsc :a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>…</mo> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(a_{1}:\dotsc :a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2119bda5c32c2965173974f49648064e39303ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.983ex; height:2.843ex;" alt="{\displaystyle P=(a_{1}:\dotsc :a_{n})}">. Für die Ecken gilt offensichtlich <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;X_{1}=(1:0:0:\ldots :0),\;X_{2}=(0:1:0\dotsc :0)\dotsc ,\;X_{n}=(0:0:0:\dotsc :1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mo>…</mo> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>…</mo> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mo>…</mo> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;X_{1}=(1:0:0:\ldots :0),\;X_{2}=(0:1:0\dotsc :0)\dotsc ,\;X_{n}=(0:0:0:\dotsc :1)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d886c2fb579ed6bb50edfdf20859a5a2f416cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.784ex; height:2.843ex;" alt="{\displaystyle \;X_{1}=(1:0:0:\ldots :0),\;X_{2}=(0:1:0\dotsc :0)\dotsc ,\;X_{n}=(0:0:0:\dotsc :1)\;}">.</dd></dl> Baryzentrische Koordinaten sind nicht eindeutig: Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> ungleich Null beschreibt auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda a_{1}:\dotsc :\lambda a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>…</mo> <mo>:</mo> <mi>λ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda a_{1}:\dotsc :\lambda a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1194d3679772df37e43b9c48fa64898886f7c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.85ex; height:2.843ex;" alt="{\displaystyle (\lambda a_{1}:\dotsc :\lambda a_{n})}"> den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">. D. h.: Nur die Verhältnisse der Koordinaten sind wesentlich. An diese Eigenschaft soll die Schreibweise mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> erinnern. Man kann baryzentrische Koordinaten als <a href="/wiki/Homogene_Koordinaten" title="Homogene Koordinaten">homogene Koordinaten</a> eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}">-dimensionalen projektiven Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"> auffassen, von dem der affine Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"> ein Teil ist. Und zwar sind die Punkte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"> diejenigen Punkte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}">, die nicht in der durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;a_{1}+\dotsb +a_{n}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;a_{1}+\dotsb +a_{n}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b639ce9a836dbf472d90afaf9e70cb2aa263db74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.688ex; height:2.509ex;" alt="{\displaystyle \;a_{1}+\dotsb +a_{n}=0\;}"> bestimmten <a href="/wiki/Hyperebene" title="Hyperebene">Hyperebene</a> (Fernhyperebene) liegen. Gleichung (G) ist ein unterbestimmtes homogenes lineares Gleichungssystem, das sich in der üblichen Form <dl><dd>(G')<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad a_{1}(\mathbf {p} -\mathbf {x} _{1})+\dotsb +a_{n}(\mathbf {p} -\mathbf {x} _{n})=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad a_{1}(\mathbf {p} -\mathbf {x} _{1})+\dotsb +a_{n}(\mathbf {p} -\mathbf {x} _{n})=\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842325d26dc6ee2378214dd480c2b78064524699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.259ex; height:2.843ex;" alt="{\displaystyle \quad a_{1}(\mathbf {p} -\mathbf {x} _{1})+\dotsb +a_{n}(\mathbf {p} -\mathbf {x} _{n})=\mathbf {0} }"></dd></dl> schreiben lässt. Erfüllen die Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dotsc a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dotsc a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c26de2632989d0353c0326d474454e6ced7fd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.877ex; height:2.009ex;" alt="{\displaystyle a_{1},\dotsc a_{n}}"> zusätzlich die Normierungsbedingung <dl><dd>(N) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad a_{1}+\dotsb +a_{n}=1\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad a_{1}+\dotsb +a_{n}=1\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6870d07e3c38a53774522bd3cf96e385fe480c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.947ex; height:2.509ex;" alt="{\displaystyle \quad a_{1}+\dotsb +a_{n}=1\ ,}"></dd></dl> so spricht man von normierten baryzentrischen Koordinaten. In diesem Fall sind die Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}"> eindeutig bestimmt (s. unten) und man kann den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> (Ursprungsgerade) auch als affinen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\dots ,a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f32ea8be17cff0b2d07309b931aa8a6af949ac8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle (a_{1},\dots ,a_{n})}"> der Hyperebene des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> mit der Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+\dots +a_{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+\dots +a_{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b698cd2b41b8eb359a4ba565d2f7e0de23559c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.397ex; height:2.509ex;" alt="{\displaystyle a_{1}+\dots +a_{n}=1}"> auffassen. Um die Normierung formal sicherzustellen, kann man (N) nach einer Koordinate auflösen und in das n-tupel einfügen. Löst man z. B. nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"> auf, ergibt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(a_{1}:\dots :1-a_{1}-\dots -a_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(a_{1}:\dots :1-a_{1}-\dots -a_{n-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d19c0d4ef9ef47d69d9608272db4a84944272a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.775ex; height:2.843ex;" alt="{\displaystyle P=(a_{1}:\dots :1-a_{1}-\dots -a_{n-1})}">. Hinweis: Die Begriffe werden nicht einheitlich verwendet. Viele Autoren sprechen nur dann von baryzentrischen Koordinaten, wenn die Normierungsbedingung erfüllt ist. Normierte baryzentrische Koordinaten lassen sich einfach ermitteln, indem man jede einzelne baryzentrische Koordinate durch die Summe der Koordinaten dividiert. <div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=3" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Punkt_im_Simplex">Punkt im Simplex</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=4" title="Abschnitt bearbeiten: Punkt im Simplex" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Punkt im Simplex">Quelltext bearbeiten</a>]</div> Falls die Koordinaten positiv sind, so liegt der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> in der <a href="/wiki/Konvexe_H%C3%BClle" title="Konvexe Hülle">konvexen Hülle</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dotsc ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25602d4b1df2584ef4dd94474b2db32c4e970e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}}">, also im Simplex mit diesen Eckpunkten. Die Darstellung eines Punktes innerhalb einer konvexen Hülle als Summe von Eckpunkten eines Simplex wird affine Kombination oder baryzentrische Kombination genannt. <div class="mw-heading mw-heading4"><h4 id="Massenmittelpunkt">Massenmittelpunkt</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=5" title="Abschnitt bearbeiten: Massenmittelpunkt" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Massenmittelpunkt">Quelltext bearbeiten</a>]</div> Wie man aus der Umstellung <dl><dd>(S)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mathbf {p} ={\frac {a_{1}\mathbf {x} _{1}+\dotsb +a_{n}\mathbf {x} _{n}}{a_{1}+\dotsb +a_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {p} ={\frac {a_{1}\mathbf {x} _{1}+\dotsb +a_{n}\mathbf {x} _{n}}{a_{1}+\dotsb +a_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab327a83130c5fcb0435b614ac61fe342c656b17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.973ex; height:5.343ex;" alt="{\displaystyle \quad \mathbf {p} ={\frac {a_{1}\mathbf {x} _{1}+\dotsb +a_{n}\mathbf {x} _{n}}{a_{1}+\dotsb +a_{n}}}}"></dd></dl> der Definitionsgleichung (G) sieht, kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> als <a href="/wiki/Massenmittelpunkt" title="Massenmittelpunkt">Massenmittelpunkt</a> (das <a href="/wiki/Baryzentrum" title="Baryzentrum">Baryzentrum</a>) einer Anordnung von Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dotsc ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dotsc ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bda393afa1b6471d6a48ff74e876d1d02f4415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dotsc ,a_{n}}"> an den Eckpunkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dotsc ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25602d4b1df2584ef4dd94474b2db32c4e970e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}}"> des Simplex auffassen. Dies ist der Ursprung des Begriffs baryzentrisch. Physikalische Bedeutung der <ul><li>Gleichung (G): Die Gesamtmasse im Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> verursacht im Nullpunkt dasselbe Drehmoment wie die Einzelmassen,</li> <li>Gleichung (G'): Die Summe der von den Einzelmassen erzeugten Drehmomente ist im Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> gleich 0.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Mittelpunkt_zweier_Punkte">Mittelpunkt zweier Punkte</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=6" title="Abschnitt bearbeiten: Mittelpunkt zweier Punkte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Mittelpunkt zweier Punkte">Quelltext bearbeiten</a>]</div> Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{1}:\dots :p_{n}),(q_{1}:\dots :q_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{1}:\dots :p_{n}),(q_{1}:\dots :q_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd0a789e3bb01c99d69450c2428658800863059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.806ex; height:2.843ex;" alt="{\displaystyle (p_{1}:\dots :p_{n}),(q_{1}:\dots :q_{n})}"> die normierten (!) baryzentrischen Darstellungen zweier Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:\mathbf {p} ,Q:\mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mi>Q</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P:\mathbf {p} ,Q:\mathbf {q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f08b2157367ab8ee5a2a3100137ccbe86453fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.393ex; height:2.509ex;" alt="{\displaystyle P:\mathbf {p} ,Q:\mathbf {q} }">, dann hat der Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:{\tfrac {1}{2}}(\mathbf {p} +\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:{\tfrac {1}{2}}(\mathbf {p} +\mathbf {q} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473d76962d2cad46ca6878ffb783fa998471d260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.588ex; height:3.509ex;" alt="{\displaystyle M:{\tfrac {1}{2}}(\mathbf {p} +\mathbf {q} )}"> die baryzentrische Darstellung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\left({\frac {p_{1}+q_{1}}{2}}:\dots :{\frac {p_{n}+q_{n}}{2}}\right)=(p_{1}+q_{1}:\dots :p_{n}+q_{n})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\left({\frac {p_{1}+q_{1}}{2}}:\dots :{\frac {p_{n}+q_{n}}{2}}\right)=(p_{1}+q_{1}:\dots :p_{n}+q_{n})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bc86e317ecdf0d30a875eca9c2b690f9cc47a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.243ex; height:6.176ex;" alt="{\displaystyle M=\left({\frac {p_{1}+q_{1}}{2}}:\dots :{\frac {p_{n}+q_{n}}{2}}\right)=(p_{1}+q_{1}:\dots :p_{n}+q_{n})\ .}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Existenz,_Eindeutigkeit_normierter_Koordinaten">Existenz, Eindeutigkeit normierter Koordinaten</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=7" title="Abschnitt bearbeiten: Existenz, Eindeutigkeit normierter Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Existenz, Eindeutigkeit normierter Koordinaten">Quelltext bearbeiten</a>]</div> Normierte baryzentrische Koordinaten sind eindeutig bestimmt. Denn, versucht man das durch (G') und (N) beschriebene inhomogene <a href="/wiki/Lineares_Gleichungssystem" title="Lineares Gleichungssystem">lineare Gleichungssystem</a> mit Hilfe der <a href="/wiki/Cramersche_Regel" title="Cramersche Regel">Cramerschen Regel</a> zu lösen, ist die Determinante im Nenner ungleich Null, da sie, bis auf einen Faktor, im ebenen Fall (n=3) die orientierte Fläche des Dreiecks und im 3-dimensionalen Fall (n=4) das orientierte Volumen des Tetraeders ist (siehe unten). Lässt man die Bedingung (N) wieder fallen, hat das lineare homogene System (G') 1-dimensionale Lösungen (Punkte des oben erwähnten projektiven Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}">). Für größeres <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> gilt Entsprechendes. <div class="mw-heading mw-heading4"><h4 id="Unabhängigkeit_von_Nullpunkt_und_Skalierung">Unabhängigkeit von Nullpunkt und Skalierung</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=8" title="Abschnitt bearbeiten: Unabhängigkeit von Nullpunkt und Skalierung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Unabhängigkeit von Nullpunkt und Skalierung">Quelltext bearbeiten</a>]</div> Dass die baryzentrischen Koordinaten nicht von dem zufällig gewählten Nullpunkt des affinen Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"> abhängen, erkennt man dadurch, dass eine Verschiebung der Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ,\mathbf {x} _{1},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ,\mathbf {x} _{1},\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c576d71bb0aaf9dfaa844d7ef2687e83c5ef8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.742ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} ,\mathbf {x} _{1},\dots }"> um einen festen Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> die Definitionsgleichung (G) unverändert lässt. Dasselbe gilt für eine uniforme Skalierung (Multiplikation der Vektoren mit einem festen Faktor ungleich Null). <div class="mw-heading mw-heading4"><h4 id="Beispiel">Beispiel</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=9" title="Abschnitt bearbeiten: Beispiel" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Beispiel">Quelltext bearbeiten</a>]</div> In der Ebene besteht ein Simplex aus 3 Punkten (Dreieck), d. h. es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"> und jeder Punkt hat 3 baryzentrische Koordinaten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(a_{1}:a_{2}:a_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(a_{1}:a_{2}:a_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985ac802875924c89042a2ab494754a8ce02ab54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.38ex; height:2.843ex;" alt="{\displaystyle P=(a_{1}:a_{2}:a_{3})}">. Zum Beispiel hat der <a href="/wiki/Geometrischer_Schwerpunkt" title="Geometrischer Schwerpunkt">geometrische Schwerpunkt</a> des Dreiecks die baryzentrische Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;S=(1:1:1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;S=(1:1:1)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071b3ad46feee291bca10ea625857274b32db9ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.059ex; height:2.843ex;" alt="{\displaystyle \;S=(1:1:1)\;}">, denn es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathbf {s} ={\tfrac {1}{3}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3})\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathbf {s} ={\tfrac {1}{3}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3})\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a13f95a36bd54eda2a961e6c05ea012a4e344d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.635ex; height:3.676ex;" alt="{\displaystyle \;\mathbf {s} ={\tfrac {1}{3}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3})\;.}"> Die normierte Darstellung ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;S=({\tfrac {1}{3}}:{\tfrac {1}{3}}:{\tfrac {1}{3}})\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;S=({\tfrac {1}{3}}:{\tfrac {1}{3}}:{\tfrac {1}{3}})\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e829c852dda26b0f06fd2bd41abe858ccd455f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.193ex; height:3.676ex;" alt="{\displaystyle \;S=({\tfrac {1}{3}}:{\tfrac {1}{3}}:{\tfrac {1}{3}})\;.}"> <div class="mw-heading mw-heading4"><h4 id="Vorteil,_Nachteil">Vorteil, Nachteil</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=10" title="Abschnitt bearbeiten: Vorteil, Nachteil" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Vorteil, Nachteil">Quelltext bearbeiten</a>]</div> Wie man in dem Beispiel sieht, lassen sich wesentliche Punkte z. B. von Dreiecken einheitlich und einfach beschreiben. Bei Berechnungen müssen nicht die speziellen (affinen) Koordinaten eines gegebenen Dreiecks berücksichtigt werden. Wie man affine Koordinaten in baryzentrische Koordinaten umrechnet, wird in den folgenden Abschnitten gezeigt. Ein gewisser Nachteil baryzentrischer Koordinaten ist allerdings: Sie sind nicht eindeutig (im nicht normierten Fall) und es gibt immer 1 Koordinate mehr als die affinen Koordinaten. <div class="mw-heading mw-heading4"><h4 id="Unterschied_zu_anderen_homogenen_Koordinaten:_Beispiel_n=3">Unterschied zu anderen homogenen Koordinaten: Beispiel n=3</h4>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=11" title="Abschnitt bearbeiten: Unterschied zu anderen homogenen Koordinaten: Beispiel n=3" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Unterschied zu anderen homogenen Koordinaten: Beispiel n=3">Quelltext bearbeiten</a>]</div> Üblicherweise führt man homogene Koordinaten so ein, dass die Ferngerade durch eine Koordinatenebene, z. B. durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a931032fea4e053b3719d07cf2abac0b18a7f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{3}=0}">, beschrieben wird. Dies hat den Vorteil, dass ein einfacher Zusammenhang zu den affinen Koordinaten, die die zugehörige affine Ebene (projektive Ebene ohne die Punkte der Ferngerade) beschreiben, besteht: Ein affiner Punkt hat die Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})=(a_{1}:a_{2}:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})=(a_{1}:a_{2}:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda4503fda88be80a5155f743c6dbd9617a4aef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.924ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})=(a_{1}:a_{2}:1)}">. Es besteht allerdings der Nachteil, dass die zu den Koordinatenachsen gehörigen projektiven Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:0:0),(0:1:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:0:0),(0:1:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83329f8b060f3cb3d5ec77cabc0c39f8c5438c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.376ex; height:2.843ex;" alt="{\displaystyle (1:0:0),(0:1:0)}"> keine affinen Punkte sind. Nur der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0:0:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0:0:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1524ca8de0b50e486a82ff3676d0c8ad4a7825db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle (0:0:1)}"> wird zu einem affinen Punkt. Baryzentrische Koordinaten haben keine so einfache Beziehung zu den affinen Koordinaten. Dafür liegen alle den Koordinatenachsen entsprechenden projektiven Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:0:0),(0:1:0),(0:0:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:0:0),(0:1:0),(0:0:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfd776344a23752ebf036fdd7f06ae796475f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.581ex; height:2.843ex;" alt="{\displaystyle (1:0:0),(0:1:0),(0:0:1)}"> im affinen Bereich, denn die Ferngerade wird hier durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+a_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+a_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74e33ad9a266e258440985b80d36bd01ea31e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.794ex; height:2.509ex;" alt="{\displaystyle a_{1}+a_{2}+a_{3}=0}"> beschrieben. <div class="mw-heading mw-heading2"><h2 id="Auf_einer_Gerade_(n=2,_Strecke)">Auf einer Gerade (n=2, Strecke)</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=12" title="Abschnitt bearbeiten: Auf einer Gerade (n=2, Strecke)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Auf einer Gerade (n=2, Strecke)">Quelltext bearbeiten</a>]</div> Der Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1b40229a003b8b719a29ac70c15fe4f704a777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}"> zweier Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd2f8098df7628f42d6b2f8fdb7450122aefec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.223ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2}}">, die auf der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse an den Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188263943645114e27e316cc1be787861e5b67be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.802ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2}}"> platziert sind, ist die Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731c17e8ce0dad32aab6e3ad68f52405fe277007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.333ex; height:2.009ex;" alt="{\displaystyle x_{s}}">, wo das <a href="/wiki/Hebelgesetz" class="mw-redirect" title="Hebelgesetz">Hebelgesetz</a> (Kraft × Kraftarm = Last × Lastarm, siehe 2. Bild) erfüllt ist. Genauer: Wo die Summe der <a href="/wiki/Drehmoment" title="Drehmoment">Drehmomente</a> gleich Null ist<a href="#cite_note-5">[5]</a> und damit gilt: <dl><dd>(G'2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa5197371b8e4f6e2f42d014e44fe2670c38e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.604ex; height:2.843ex;" alt="{\displaystyle \ m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})=0}"></dd></dl> Diese Gleichung ist äquivalent zu (siehe Abschnitt Definition) <dl><dd>(G2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (m_{1}+m_{2})x_{s}=m_{1}x_{1}+m_{2}x_{2}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (m_{1}+m_{2})x_{s}=m_{1}x_{1}+m_{2}x_{2}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04e2b87d1fa0ddafbe2841640aebf72800e4547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.941ex; height:2.843ex;" alt="{\displaystyle \ (m_{1}+m_{2})x_{s}=m_{1}x_{1}+m_{2}x_{2}\;.}"></dd></dl> Auflösen nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731c17e8ce0dad32aab6e3ad68f52405fe277007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.333ex; height:2.009ex;" alt="{\displaystyle x_{s}}"> ergibt: <dl><dd>(S2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{s}={\frac {m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{s}={\frac {m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7889ce7f7501577b1b1bccdab1dd9e40abe9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.646ex; height:5.343ex;" alt="{\displaystyle \ x_{s}={\frac {m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}}}"></dd></dl> Lässt man negative Massen zu, z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}=1,m_{2}=-1+{\tfrac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}=1,m_{2}=-1+{\tfrac {1}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abff31cfb286890fb1264a60a16f152434bd6c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.216ex; height:3.343ex;" alt="{\displaystyle m_{1}=1,m_{2}=-1+{\tfrac {1}{n}}}">, so ergibt sich aus (G2) für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"> die Gesamtmasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83d9b7fca4fa06e928961412c2ac9176ae8d33b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.291ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}=0}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{s}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{s}=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49dc95dc638aff2ba1a24db1a02724cf4bdd1b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.755ex; height:2.009ex;" alt="{\displaystyle x_{s}=\infty }">. Eine Lösung von (G'2) ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;m_{1}=x_{2}-x_{s},\;m_{2}=x_{s}-x_{1}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m_{1}=x_{2}-x_{s},\;m_{2}=x_{s}-x_{1}\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1339e722813cefc1b4307215669aecb524a5d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.47ex; height:2.343ex;" alt="{\displaystyle \;m_{1}=x_{2}-x_{s},\;m_{2}=x_{s}-x_{1}\;}">. Alle Lösungen sind Vielfache davon. Also hat der Schwerpunkt die baryzentrische Darstellung (siehe Abschnitt Definition) <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-gerade-l1l2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryzentr-ko-gerade-l1l2.svg/260px-Baryzentr-ko-gerade-l1l2.svg.png" decoding="async" width="260" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryzentr-ko-gerade-l1l2.svg/390px-Baryzentr-ko-gerade-l1l2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryzentr-ko-gerade-l1l2.svg/520px-Baryzentr-ko-gerade-l1l2.svg.png 2x" data-file-width="260" data-file-height="62" /></a><figcaption>Baryzentrische Koordinaten als Verhältnis von Strecken</figcaption></figure> <dl><dd>(B2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;X_{s}=(m_{1}:m_{2})=(x_{2}-x_{s}:x_{s}-x_{1})=(l_{\color {red}2}:l_{\color {red}1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>2</mn> </mstyle> </mrow> </msub> <mo>:</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;X_{s}=(m_{1}:m_{2})=(x_{2}-x_{s}:x_{s}-x_{1})=(l_{\color {red}2}:l_{\color {red}1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4bbe1f3e5591871f89fd5a590c12bdf0aa8ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.907ex; height:2.843ex;" alt="{\displaystyle \;X_{s}=(m_{1}:m_{2})=(x_{2}-x_{s}:x_{s}-x_{1})=(l_{\color {red}2}:l_{\color {red}1})}"></dd></dl> Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;l_{1}=x_{s}-x_{1},\;l_{2}=x_{2}-x_{s}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;l_{1}=x_{s}-x_{1},\;l_{2}=x_{2}-x_{s}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5300a9fcfe4ae451b69b6989f4a67bee778f16dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.423ex; height:2.509ex;" alt="{\displaystyle \;l_{1}=x_{s}-x_{1},\;l_{2}=x_{2}-x_{s}\;.}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-gerade.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Baryzentr-ko-gerade.svg/440px-Baryzentr-ko-gerade.svg.png" decoding="async" width="440" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Baryzentr-ko-gerade.svg/660px-Baryzentr-ko-gerade.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Baryzentr-ko-gerade.svg/880px-Baryzentr-ko-gerade.svg.png 2x" data-file-width="447" data-file-height="97" /></a><figcaption>Baryzentrische Koordinaten auf einer Gerade (unten). Der Mittelpunkt der Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6099d6fb3c34ad5e22fad9c79c40c4ebfee1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}}"> hat die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:1)=({\tfrac {1}{2}}:{\tfrac {1}{2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:1)=({\tfrac {1}{2}}:{\tfrac {1}{2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e05a6387cdaf9c1f3c083991802b719869edccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.233ex; height:3.509ex;" alt="{\displaystyle (1:1)=({\tfrac {1}{2}}:{\tfrac {1}{2}})}"></figcaption></figure> Dieser einfache Zusammenhang der baryzentrischen Koordinaten mit Verhältnissen von Teilstrecken ist der Grund für ihre Bedeutung in der Dreiecksgeometrie. Die Aussage (B2) ist der Lehrsatz in §21, S. 25, des Buches von Möbius. Die normierten baryzentrischen Koordinaten müssen zusätzlich zu (G'2) die Bedingung <dl><dd>(N2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad m_{1}+m_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad m_{1}+m_{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc2eb4a41c3366242cec7a7c26727021c52e785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.613ex; height:2.509ex;" alt="{\displaystyle \quad m_{1}+m_{2}=1}"></dd></dl> erfüllen. Löst man das inhomogene Gleichungssystem bestehend aus den Gleichungen (G'2), (N2) mit Hilfe der Cramerschen Regel, ergibt sich die normierte Darstellung <dl><dd>(NB2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X_{s}=\left({\frac {x_{2}-x_{s}}{x_{2}-x_{1}}}\;:\;{\frac {x_{s}-x_{1}}{x_{2}-x_{1}}}\right)=\left({\frac {l_{2}}{l_{1}+l_{2}}}:{\frac {l_{1}}{l_{1}+l_{2}}}\right)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X_{s}=\left({\frac {x_{2}-x_{s}}{x_{2}-x_{1}}}\;:\;{\frac {x_{s}-x_{1}}{x_{2}-x_{1}}}\right)=\left({\frac {l_{2}}{l_{1}+l_{2}}}:{\frac {l_{1}}{l_{1}+l_{2}}}\right)\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c410500d25114a7702e476be4f6ce65d4d790bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.623ex; height:6.176ex;" alt="{\displaystyle \ X_{s}=\left({\frac {x_{2}-x_{s}}{x_{2}-x_{1}}}\;:\;{\frac {x_{s}-x_{1}}{x_{2}-x_{1}}}\right)=\left({\frac {l_{2}}{l_{1}+l_{2}}}:{\frac {l_{1}}{l_{1}+l_{2}}}\right)\;.}"></dd></dl> Beispiel: Der Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {x_{1}+x_{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {x_{1}+x_{2}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303a3ff5e89798c879b03272d2234e7296de9a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {x_{1}+x_{2}}{2}}}"> der Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188263943645114e27e316cc1be787861e5b67be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.802ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2}}"> besitzt die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9439f6e87539371c87c09b033ae9a00be87b9257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.071ex; height:2.843ex;" alt="{\displaystyle (1:1)}"> und in normierter Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {1}{2}}:{\tfrac {1}{2}})\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\tfrac {1}{2}}:{\tfrac {1}{2}})\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ced6e370590aafdc2ec71fac241ea1eec43b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.355ex; height:3.509ex;" alt="{\displaystyle ({\tfrac {1}{2}}:{\tfrac {1}{2}})\;.}"> <div class="mw-heading mw-heading2"><h2 id="In_einer_Ebene_(n=3,_Dreieck)">In einer Ebene (n=3, Dreieck)</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=13" title="Abschnitt bearbeiten: In einer Ebene (n=3, Dreieck)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: In einer Ebene (n=3, Dreieck)">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Umrechnung_der_Koordinaten">Umrechnung der Koordinaten</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=14" title="Abschnitt bearbeiten: Umrechnung der Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Umrechnung der Koordinaten">Quelltext bearbeiten</a>]</div> Sind in den Ecken eines Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;X_{1}=(x_{1},y_{1}),\;X_{2}=(x_{2},y_{2}),\;X_{3}=(x_{3},y_{3}),\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;X_{1}=(x_{1},y_{1}),\;X_{2}=(x_{2},y_{2}),\;X_{3}=(x_{3},y_{3}),\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae62f58044019b979b76a219df61fcc4932a10cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.788ex; height:2.843ex;" alt="{\displaystyle \;X_{1}=(x_{1},y_{1}),\;X_{2}=(x_{2},y_{2}),\;X_{3}=(x_{3},y_{3}),\;}"> drei Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},m_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},m_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3964c8bc26f931bf70ee2d8bd12a266f2e817c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.352ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},m_{3}}"> platziert, so sind die Gleichgewichtsgleichungen für die Drehmomente um die Koordinatenachsen <dl><dd>(G'3)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\begin{array}{r}m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})+m_{3}(x_{s}-x_{3})=0\\m_{1}(y_{s}-y_{1})+m_{2}(y_{s}-y_{2})+m_{3}(y_{s}-y_{3})=0\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\begin{array}{r}m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})+m_{3}(x_{s}-x_{3})=0\\m_{1}(y_{s}-y_{1})+m_{2}(y_{s}-y_{2})+m_{3}(y_{s}-y_{3})=0\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b4397fdac2880f88632f97e1e9d7019a80830a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.4ex; height:6.176ex;" alt="{\displaystyle \quad {\begin{array}{r}m_{1}(x_{s}-x_{1})+m_{2}(x_{s}-x_{2})+m_{3}(x_{s}-x_{3})=0\\m_{1}(y_{s}-y_{1})+m_{2}(y_{s}-y_{2})+m_{3}(y_{s}-y_{3})=0\end{array}}}"></dd></dl> oder in der Form (siehe Definition) <dl><dd>(G3)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\begin{array}{c}(m_{1}+m_{2}+m_{3})x_{s}=m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}\\(m_{1}+m_{2}+m_{3})y_{s}=m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\begin{array}{c}(m_{1}+m_{2}+m_{3})x_{s}=m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}\\(m_{1}+m_{2}+m_{3})y_{s}=m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64dbd8ebfd03e7519748c15c5f143623ca58e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.396ex; height:6.176ex;" alt="{\displaystyle \quad {\begin{array}{c}(m_{1}+m_{2}+m_{3})x_{s}=m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}\\(m_{1}+m_{2}+m_{3})y_{s}=m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}\end{array}}}"></dd></dl> Der Schwerpunkt hat die Koordinaten <dl><dd>(S3)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\begin{array}{c}x_{s}=\displaystyle {\frac {m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}}{m_{1}+m_{2}+m_{3}}}\\y_{s}=\displaystyle {\frac {m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}}.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\begin{array}{c}x_{s}=\displaystyle {\frac {m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}}{m_{1}+m_{2}+m_{3}}}\\y_{s}=\displaystyle {\frac {m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}}.\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b900b606a71f8fd1dffa1c9f2bbc945b974a2472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:31.458ex; height:11.509ex;" alt="{\displaystyle \quad {\begin{array}{c}x_{s}=\displaystyle {\frac {m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}}{m_{1}+m_{2}+m_{3}}}\\y_{s}=\displaystyle {\frac {m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}}.\end{array}}}"></dd></dl> Baryzentrische Koordinaten eines gegebenen Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=({\color {red}x_{s}},{\color {red}y_{s}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=({\color {red}x_{s}},{\color {red}y_{s}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9733a960b6010be5e853af72997da7de178fc859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.916ex; height:2.843ex;" alt="{\displaystyle S=({\color {red}x_{s}},{\color {red}y_{s}})}">, erhält man durch Lösen des unterbestimmten homogenen Systems (G'3) nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},m_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},m_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3964c8bc26f931bf70ee2d8bd12a266f2e817c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.352ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},m_{3}}">. Nimmt man die Normierungsgleichung <dl><dd><dl><dd>(N3) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a05300c940820c44ad8dbf8bd5ae76d7bf72a31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.548ex; height:2.509ex;" alt="{\displaystyle \quad m_{1}+m_{2}+m_{3}=1}"></dd></dl></dd> <dd>hinzu, ist das jetzt inhomogene LGS eindeutig und mit Hilfe der Cramerschen Regel lösbar. Es ergibt sich:</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-3eck.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Baryzentr-ko-3eck.svg/220px-Baryzentr-ko-3eck.svg.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Baryzentr-ko-3eck.svg/330px-Baryzentr-ko-3eck.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Baryzentr-ko-3eck.svg/440px-Baryzentr-ko-3eck.svg.png 2x" data-file-width="186" data-file-height="145" /></a><figcaption><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(x_{s},y_{s}),\;X_{i}=(x_{i},y_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(x_{s},y_{s}),\;X_{i}=(x_{i},y_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6bcc688b20cf31d210105f04308e20660954563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.33ex; height:2.843ex;" alt="{\displaystyle S=(x_{s},y_{s}),\;X_{i}=(x_{i},y_{i})}"></figcaption></figure> <dl><dd><dl><dd>(NB3) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad {\begin{array}{l}m_{1}=\displaystyle {\frac {(x_{2}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})-(x_{3}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{2}=\displaystyle {\frac {(x_{3}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})-(x_{1}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{3}=\displaystyle {\frac {(x_{1}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})-(x_{2}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad {\begin{array}{l}m_{1}=\displaystyle {\frac {(x_{2}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})-(x_{3}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{2}=\displaystyle {\frac {(x_{3}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})-(x_{1}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{3}=\displaystyle {\frac {(x_{1}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})-(x_{2}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80bc5588e7d8f75fb1329d76a1a00af74c1a9813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:52.175ex; height:20.176ex;" alt="{\displaystyle \qquad {\begin{array}{l}m_{1}=\displaystyle {\frac {(x_{2}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})-(x_{3}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{2}=\displaystyle {\frac {(x_{3}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})-(x_{1}-{\color {red}x_{s}})(y_{3}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\m_{3}=\displaystyle {\frac {(x_{1}-{\color {red}x_{s}})(y_{2}-{\color {red}y_{s}})-(x_{2}-{\color {red}x_{s}})(y_{1}-{\color {red}y_{s}})}{(x_{2}-x_{1})(y_{3}-y_{2})-(y_{2}-y_{1})(x_{3}-x_{2})}}\\\end{array}}}"></dd></dl></dd> <dd>Der gemeinsame Nenner ist der doppelte Flächeninhalt des Dreiecks, also ungleich Null.</dd> <dd>Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa64a502bfee18a11f4430e03682542f1cc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.226ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> genügt es, zwei der drei Brüche zu berechnen.</dd> <dd>Alle Zähler lassen sich als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}">-Determinanten schreiben. Verzichtet man auf die Normierung, darf bei den baryzentrischen Koordinaten der gemeinsame Nenner weggelassen werden:</dd></dl> <dl><dd>(B3)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad (m_{1}:m_{2}:m_{3})={\Big (}\left|{\begin{array}{l}x_{2}\!-\!{\color {red}x_{s}}&x_{3}\!-\!{\color {red}x_{s}}\\y_{2}\!-\!{\color {red}y_{s}}&y_{3}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{3}\!-\!{\color {red}x_{s}}&x_{1}\!-\!{\color {red}x_{s}}\\y_{3}\!-\!{\color {red}y_{s}}&y_{1}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{1}\!-\!{\color {red}x_{s}}&x_{2}\!-\!{\color {red}x_{s}}\\y_{1}\!-\!{\color {red}y_{s}}&y_{2}\!-\!{\color {red}y_{s}}\\\end{array}}\right|{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad (m_{1}:m_{2}:m_{3})={\Big (}\left|{\begin{array}{l}x_{2}\!-\!{\color {red}x_{s}}&x_{3}\!-\!{\color {red}x_{s}}\\y_{2}\!-\!{\color {red}y_{s}}&y_{3}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{3}\!-\!{\color {red}x_{s}}&x_{1}\!-\!{\color {red}x_{s}}\\y_{3}\!-\!{\color {red}y_{s}}&y_{1}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{1}\!-\!{\color {red}x_{s}}&x_{2}\!-\!{\color {red}x_{s}}\\y_{1}\!-\!{\color {red}y_{s}}&y_{2}\!-\!{\color {red}y_{s}}\\\end{array}}\right|{\Big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb52fb48258aa193b961423ef32d78a345beccff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.615ex; height:6.176ex;" alt="{\displaystyle \quad (m_{1}:m_{2}:m_{3})={\Big (}\left|{\begin{array}{l}x_{2}\!-\!{\color {red}x_{s}}&x_{3}\!-\!{\color {red}x_{s}}\\y_{2}\!-\!{\color {red}y_{s}}&y_{3}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{3}\!-\!{\color {red}x_{s}}&x_{1}\!-\!{\color {red}x_{s}}\\y_{3}\!-\!{\color {red}y_{s}}&y_{1}\!-\!{\color {red}y_{s}}\\\end{array}}\right|:\left|{\begin{array}{l}x_{1}\!-\!{\color {red}x_{s}}&x_{2}\!-\!{\color {red}x_{s}}\\y_{1}\!-\!{\color {red}y_{s}}&y_{2}\!-\!{\color {red}y_{s}}\\\end{array}}\right|{\Big )}}"></dd></dl> <dl><dd>Multipliziert man jede Determinante mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}">, entstehen die orientierten <a href="/wiki/Dreiecksfl%C3%A4che" title="Dreiecksfläche">Flächen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18731d8b8a4b3883417c1b9cf18d63572ddcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.038ex; height:2.509ex;" alt="{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}"> der Teildreiecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}X_{3}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}X_{3}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd72fff3fa60547e88e56eef1f3d86a353d9479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.456ex; height:2.509ex;" alt="{\displaystyle X_{2}X_{3}S}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}X_{1}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}X_{1}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85157599569c2c757a3127b22b4bd022b1390416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.456ex; height:2.509ex;" alt="{\displaystyle X_{3}X_{1}S}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967006866e464906c3090ff135cc7817ebab4119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.456ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}S}"> (siehe auch den nächsten Abschnitt Beziehung zu trilineare Koordinaten). Damit gilt:</dd> <dd>(BF3)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad (m_{1}:m_{2}:m_{3})=(\Delta _{1}:\Delta _{2}:\Delta _{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad (m_{1}:m_{2}:m_{3})=(\Delta _{1}:\Delta _{2}:\Delta _{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c651f532056501f02afc1e5abbb165320db28c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.365ex; height:2.843ex;" alt="{\displaystyle \qquad (m_{1}:m_{2}:m_{3})=(\Delta _{1}:\Delta _{2}:\Delta _{3})}"></dd></dl> Aussage (BF3) ist der Lehrsatz in §23, S. 26, des Buches von Möbius. Spezialfall: Koordinatendreieck: Für das spezielle rechtwinklige Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;X_{3}=(0,0),X_{1}=(1,0),X_{2}=(0,1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;X_{3}=(0,0),X_{1}=(1,0),X_{2}=(0,1)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19817f5c32b3464e8247c81dd150ae5cafbdd4de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.094ex; height:2.843ex;" alt="{\displaystyle \;X_{3}=(0,0),X_{1}=(1,0),X_{2}=(0,1)\;}"> als Bezugsdreieck hat ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> die einfachen baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x:y:1-x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mi>y</mi> <mo>:</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x:y:1-x-y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda3031635bf1c3c299eee662c95dfa7b4bafe09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.497ex; height:2.843ex;" alt="{\displaystyle (x:y:1-x-y)}">. <div class="mw-heading mw-heading3"><h3 id="Geraden,_Schnittpunkte,_Parallelität">Geraden, Schnittpunkte, Parallelität</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=15" title="Abschnitt bearbeiten: Geraden, Schnittpunkte, Parallelität" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Geraden, Schnittpunkte, Parallelität">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-ebene.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Baryzentr-ko-ebene.svg/350px-Baryzentr-ko-ebene.svg.png" decoding="async" width="350" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Baryzentr-ko-ebene.svg/525px-Baryzentr-ko-ebene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Baryzentr-ko-ebene.svg/700px-Baryzentr-ko-ebene.svg.png 2x" data-file-width="457" data-file-height="307" /></a><figcaption>In den Punkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4126aa4015fd02d5478fd2d09601afd33286509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}}"> befinden sich die Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},m_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},m_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3964c8bc26f931bf70ee2d8bd12a266f2e817c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.352ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},m_{3}}">. Die lilafarbigen parallelen Geraden haben die jeweils angegebenen Gleichungen. Ihr gemeinsamer Fernpunkt hat die Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1:1:0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1:1:0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500607e9b0381abf41c75c40650135cd690529a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.626ex; height:2.843ex;" alt="{\displaystyle (-1:1:0).}"> Die Koordinaten der Rasterpunkte sind normiert.</figcaption></figure> <ul><li>Die Ecken des Dreiecks haben die homogenen Koordinaten</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1\!:\!0\!:\!0),\;X_{2}=(0\!:\!1\!:\!0),\;X_{3}=(0\!:\!0\!:\!1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>0</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>1</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>0</mn> <mspace width="negativethinmathspace" /> <mo>:</mo> <mspace width="negativethinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1\!:\!0\!:\!0),\;X_{2}=(0\!:\!1\!:\!0),\;X_{3}=(0\!:\!0\!:\!1)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850ba7b544ee0b1e50f0ce4479291eeeecf3b51a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.102ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1\!:\!0\!:\!0),\;X_{2}=(0\!:\!1\!:\!0),\;X_{3}=(0\!:\!0\!:\!1)\;}">.</dd></dl> <ul><li>Die Gerade durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6099d6fb3c34ad5e22fad9c79c40c4ebfee1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}}"> wird durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af7213064a4a766ce7ed1e85449eb0f04f3344d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{3}=0}"> beschrieben und hat den Fernpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1:1:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1:1:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2eecc144117d67e1312e0548890828e8d24a19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle (-1:1:0)}">. …</li> <li>Die Ferngerade ist durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba4ff2bd52994388beae227fb82b7a8f92e3d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.516ex; height:2.509ex;" alt="{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}"> festgelegt.</li> <li>Eine beliebige Gerade wird durch eine Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>a</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>c</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f392fccc838eaa3bb18e6a81d17ea9774bb0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.75ex; height:2.509ex;" alt="{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}"> beschrieben (s. homogene Koordinaten).</li> <li>Drei Geraden</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ec4123a1292d1a450d2ea167f5349f29744c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.269ex; height:2.509ex;" alt="{\displaystyle a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99ee3b42fcf89af759368df2bdeae0acad90048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.269ex; height:2.509ex;" alt="{\displaystyle a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}m_{1}+b_{3}m_{2}+c_{3}m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}m_{1}+b_{3}m_{2}+c_{3}m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91cfa2fed308fce744d6a7d40b971773b7c5787c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.623ex; height:2.509ex;" alt="{\displaystyle a_{3}m_{1}+b_{3}m_{2}+c_{3}m_{3}=0}"></dd> <dd>haben einen Punkt gemeinsam, wenn</dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{matrix}}\right|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{matrix}}\right|=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabe80b0dfd3b537022debbb3adf1c34eddeb9d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.316ex; height:9.176ex;" alt="{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{matrix}}\right|=0}">. <ul><li>Zwei Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,\;a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,\;a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f1bc17771513676f228e82b9f7863df43c7c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:54.215ex; height:2.509ex;" alt="{\displaystyle \;a_{1}m_{1}+b_{1}m_{2}+c_{1}m_{3}=0,\;a_{2}m_{1}+b_{2}m_{2}+c_{2}m_{3}=0\;}"> sind parallel, wenn sie sich auf der Ferngerade schneiden, d. h., wenn</li></ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\1&1&1\end{matrix}}\right|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\1&1&1\end{matrix}}\right|=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf82b86572329da6c9cf329545cd7ac3aa9c4b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.316ex; height:9.176ex;" alt="{\displaystyle \qquad \quad \left|{\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\1&1&1\end{matrix}}\right|=0}">. <ul><li>Drei Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{1}:m_{2}:m_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{1}:m_{2}:m_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33805be288a83ef5b5c4acac3aa4a8ab429fdca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.968ex; height:2.843ex;" alt="{\displaystyle (m_{1}:m_{2}:m_{3})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m'_{1}:m'_{2}:m'_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>:</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>:</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m'_{1}:m'_{2}:m'_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a3be2e42790653339e646e6e1d9ca32c01924d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.968ex; height:3.009ex;" alt="{\displaystyle (m'_{1}:m'_{2}:m'_{3})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m''_{1}:m''_{2}:m''_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>″</mo> </msubsup> <mo>:</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>″</mo> </msubsup> <mo>:</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m''_{1}:m''_{2}:m''_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/692945439902a18f26df91e6985eddbdfb97921e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.216ex; height:3.009ex;" alt="{\displaystyle (m''_{1}:m''_{2}:m''_{3})}"> liegen genau dann auf einer Geraden, wenn</li></ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\m'_{1}&m'_{2}&m'_{3}\\m''_{1}&m''_{2}&m''_{3}\end{matrix}}\right|=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>″</mo> </msubsup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\m'_{1}&m'_{2}&m'_{3}\\m''_{1}&m''_{2}&m''_{3}\end{matrix}}\right|=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b019bd4f8d7266c35c3840d9cebfd974eb32b112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.117ex; margin-bottom: -0.221ex; width:28.099ex; height:9.843ex;" alt="{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\m'_{1}&m'_{2}&m'_{3}\\m''_{1}&m''_{2}&m''_{3}\end{matrix}}\right|=0.}"> <ul><li>Hieraus ergibt sich die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>a</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>c</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f392fccc838eaa3bb18e6a81d17ea9774bb0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.75ex; height:2.509ex;" alt="{\displaystyle \;am_{1}+bm_{2}+cm_{3}=0\;}"> einer Gerade durch zwei vorgegebene Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{1}:u_{2}:u_{3}),(v_{1}:v_{2}:v_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{1}:u_{2}:u_{3}),(v_{1}:v_{2}:v_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7568a07fa4fcfef1540f3dddaca12f3e90c94e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.099ex; height:2.843ex;" alt="{\displaystyle (u_{1}:u_{2}:u_{3}),(v_{1}:v_{2}:v_{3})}"> in Determinantenform:</li></ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\end{matrix}}\right|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\end{matrix}}\right|=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae74a71812c7b47bedb282416a789efdbdc1824b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.203ex; height:9.176ex;" alt="{\displaystyle \qquad \quad \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\end{matrix}}\right|=0}"> <div class="mw-heading mw-heading3"><h3 id="Beziehung_zu_trilinearen_Koordinaten">Beziehung zu trilinearen Koordinaten</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=16" title="Abschnitt bearbeiten: Beziehung zu trilinearen Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Beziehung zu trilinearen Koordinaten">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-Flaeche.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Baryzentr-ko-Flaeche.svg/220px-Baryzentr-ko-Flaeche.svg.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Baryzentr-ko-Flaeche.svg/330px-Baryzentr-ko-Flaeche.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Baryzentr-ko-Flaeche.svg/440px-Baryzentr-ko-Flaeche.svg.png 2x" data-file-width="221" data-file-height="149" /></a><figcaption>Grundseite und Höhe eines Teildreiecks</figcaption></figure> Für die Flächen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18731d8b8a4b3883417c1b9cf18d63572ddcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.038ex; height:2.509ex;" alt="{\displaystyle \Delta _{1},\Delta _{2},\Delta _{3}}"> der Teildreiecke in (BF3) gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{i}={\frac {1}{2}}s_{i}d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{i}={\frac {1}{2}}s_{i}d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c361560f3e60c9b0ab5ea0262acf312cda7891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.731ex; height:5.176ex;" alt="{\displaystyle \Delta _{i}={\frac {1}{2}}s_{i}d_{i}}">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i},d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i},d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ff5f26c7f72623a770dab226cbe0d529fa61f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.933ex; height:2.509ex;" alt="{\displaystyle s_{i},d_{i}}"> die Grundseiten (Seiten des Dreiecks) und die Höhen der Teildreiecke sind (siehe Bild). Also gilt <dl><dd>(BT3) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (m_{1}:m_{2}:m_{3})=(s_{1}d_{1}:s_{2}d_{2}:s_{3}d_{3})\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (m_{1}:m_{2}:m_{3})=(s_{1}d_{1}:s_{2}d_{2}:s_{3}d_{3})\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb18ba550caa2e605e5107ed45c4093c2a71a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.781ex; height:2.843ex;" alt="{\displaystyle \ (m_{1}:m_{2}:m_{3})=(s_{1}d_{1}:s_{2}d_{2}:s_{3}d_{3})\ ,}"></dd></dl> Die Beziehung (BT3) zeigt den einfachen Zusammenhang der baryzentrischen Koordinaten mit den <a href="/wiki/Trilineare_Koordinaten" title="Trilineare Koordinaten">trilinearen Koordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d_{1}:d_{2}:d_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d_{1}:d_{2}:d_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac564b18e0fa2a4293d89af349923034f27d39ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.473ex; height:2.843ex;" alt="{\displaystyle (d_{1}:d_{2}:d_{3})}"> eines Punktes. Für ein <a href="/wiki/Gleichseitiges_Dreieck" title="Gleichseitiges Dreieck">gleichseitiges Dreieck</a> sind die baryzentrischen und trilinearen Koordinaten gleich. Die Ferngerade hat in baryzentrischen Koordinaten die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba4ff2bd52994388beae227fb82b7a8f92e3d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.516ex; height:2.509ex;" alt="{\displaystyle \;m_{1}+m_{2}+m_{3}=0\;}">. In trilinearen Koordinaten ist die Gleichung noch von den Seitenlängen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> des Dreiecks abhängig: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;s_{1}d_{1}+s_{2}d_{2}+s_{3}d_{3}=0\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;s_{1}d_{1}+s_{2}d_{2}+s_{3}d_{3}=0\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccad9f3815ba21845523eec419052208982fe2b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.102ex; height:2.509ex;" alt="{\displaystyle \;s_{1}d_{1}+s_{2}d_{2}+s_{3}d_{3}=0\;.}"> <div class="mw-heading mw-heading3"><h3 id="Besondere_Punkte,_Eulergerade">Besondere Punkte, Eulergerade</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=17" title="Abschnitt bearbeiten: Besondere Punkte, Eulergerade" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Besondere Punkte, Eulergerade">Quelltext bearbeiten</a>]</div> <dl><dt>geometrischer Schwerpunkt</dt></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> ist der geometrische Schwerpunkt, wenn alle Massen gleich sind. Seine baryzentrischen Koordinaten sind also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;(1:1:1)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;(1:1:1)\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b74a84354c986ce777092ebe9476617a0fa908f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.108ex; height:2.843ex;" alt="{\displaystyle \;(1:1:1)\;.}"> Wegen (BF3) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ={\tfrac {1}{2}}s_{i}h_{i}\;,\;\Delta _{i}={\tfrac {1}{2}}s_{i}d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\tfrac {1}{2}}s_{i}h_{i}\;,\;\Delta _{i}={\tfrac {1}{2}}s_{i}d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a9b9c934800eeb0df2f02acc9706004f1119d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.436ex; height:3.509ex;" alt="{\displaystyle \Delta ={\tfrac {1}{2}}s_{i}h_{i}\;,\;\Delta _{i}={\tfrac {1}{2}}s_{i}d_{i}}"> gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \Delta _{i}={\frac {1}{3}}\Delta \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \Delta _{i}={\frac {1}{3}}\Delta \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1b29ae5574343b981d1e53725a3d897694ad74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.414ex; height:5.176ex;" alt="{\displaystyle \quad \Delta _{i}={\frac {1}{3}}\Delta \quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad d_{i}={\frac {1}{3}}h_{i}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad d_{i}={\frac {1}{3}}h_{i}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f4103e8b4da1fd55fa626b2dbd50e5e466c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.859ex; height:5.176ex;" alt="{\displaystyle \quad d_{i}={\frac {1}{3}}h_{i}\;.}"></dd></dl> (Siehe hierzu auch <a href="/wiki/Geometrischer_Schwerpunkt#Dreieck" title="Geometrischer Schwerpunkt">Geometrischer Schwerpunkt</a>.) <dl><dt>Parameterdarstellung einer Gerade</dt></dl> Eine Gerade durch zwei Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{1}:a_{2}:a_{3}),B=(b_{1}:b_{2}:b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a_{1}:a_{2}:a_{3}),B=(b_{1}:b_{2}:b_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832eb1c70fc872ed61c74e875c3471a8e1fb1835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.113ex; height:2.843ex;" alt="{\displaystyle A=(a_{1}:a_{2}:a_{3}),B=(b_{1}:b_{2}:b_{3})}"> hat für Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neq B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6345e883cc2eadba13e343c72bc906109475c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.217ex; height:2.676ex;" alt="{\displaystyle \neq B}"> die Darstellung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)={\big (}a_{1}+tb_{1}:a_{2}+tb_{2}:a_{3}+tb_{3}{\big )}\ ,\ t\in \mathbb {R} \;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)={\big (}a_{1}+tb_{1}:a_{2}+tb_{2}:a_{3}+tb_{3}{\big )}\ ,\ t\in \mathbb {R} \;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b7a88c8b32af28e8cab0a5ed68caa754fd2666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.625ex; height:3.176ex;" alt="{\displaystyle X(t)={\big (}a_{1}+tb_{1}:a_{2}+tb_{2}:a_{3}+tb_{3}{\big )}\ ,\ t\in \mathbb {R} \;.}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-proj3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Baryko-proj3.svg/220px-Baryko-proj3.svg.png" decoding="async" width="220" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Baryko-proj3.svg/330px-Baryko-proj3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Baryko-proj3.svg/440px-Baryko-proj3.svg.png 2x" data-file-width="236" data-file-height="166" /></a><figcaption>Projektion eines Punktes auf die Seite gegenüber einer Ecke</figcaption></figure> <dl><dt>Projektion auf eine Seite</dt></dl> Projiziert man einen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(\mu _{1}:\mu _{2}:\mu _{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(\mu _{1}:\mu _{2}:\mu _{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6f457c8262ebf7f3818b0efb6e9f42a83d24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.895ex; height:2.843ex;" alt="{\displaystyle P=(\mu _{1}:\mu _{2}:\mu _{3})}"> von der Ecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}=(0:0:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}=(0:0:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d299ffbe76221aebf5da4f379ebec43abca1987c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.248ex; height:2.843ex;" alt="{\displaystyle X_{3}=(0:0:1)}"> aus auf die gegenüberliegende Seite (die Gerade hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af7213064a4a766ce7ed1e85449eb0f04f3344d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{3}=0}">), so erhält man den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{3}=(\mu _{1}:\mu _{2}:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{3}=(\mu _{1}:\mu _{2}:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af12cede4c17ad8b16e57578b8b510174b0f6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.261ex; height:2.843ex;" alt="{\displaystyle Y_{3}=(\mu _{1}:\mu _{2}:0)}"> (siehe Bild). Sind die Koordinaten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> normiert, teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}Y_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}Y_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5fe4ad6e4049540605867b9d079c4569da1f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.383ex; height:2.509ex;" alt="{\displaystyle X_{3}Y_{3}}"> im <a href="/wiki/Teilverh%C3%A4ltnis" title="Teilverhältnis">Verhältnis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\mu _{3}):\mu _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\mu _{3}):\mu _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c5024b1395cc2f89678ed25d231544aa78cc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.661ex; height:2.843ex;" alt="{\displaystyle (1-\mu _{3}):\mu _{3}}">. Ist z. B. der Punkt der geometrische Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, so wird er auf die Seitenmitte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"> projiziert und teilt die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}S_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f95ff351b3c1e728dd20f4fb3be4a84abc5eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.458ex; height:2.509ex;" alt="{\displaystyle X_{3}S_{3}}"> im Verhältnis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2:1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b7f6da2bff48470ac96115b5c5d665f00b46a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.262ex; height:2.176ex;" alt="{\displaystyle 2:1}">. Entsprechendes gilt für die Projektionen von den anderen Ecken aus. <dl><dt>Inkreismittelpunkt, Ankreismittelpunkte</dt></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-3eck-ankreis.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryko-3eck-ankreis.svg/260px-Baryko-3eck-ankreis.svg.png" decoding="async" width="260" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryko-3eck-ankreis.svg/390px-Baryko-3eck-ankreis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Baryko-3eck-ankreis.svg/520px-Baryko-3eck-ankreis.svg.png 2x" data-file-width="259" data-file-height="209" /></a><figcaption>Zu Inkreismittelpunkt und Ankreismittelpunkte: Die Flächeninhalte der Dreiecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b385a4132a8bf83c6f72b0c857a84a205bf3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.936ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}X_{3}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}X_{2}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}X_{2}X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc5f1f48f4782c911caf6875b79e31db29c62ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.754ex; height:2.509ex;" alt="{\displaystyle A_{1}X_{2}X_{3}}"> haben verschiedene Vorzeichen </figcaption></figure> Für den <a href="/wiki/Inkreis" title="Inkreis">Inkreis</a> des Dreiecks gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}=r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58058fec1b509dd55be4f610eb3b5cdf27d88374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.156ex; height:2.509ex;" alt="{\displaystyle d_{i}=r}"> (Inkreisradius) und damit (s. (BT3)) hat der Inkreismittelpunkt die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{1}:s_{2}:s_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s_{1}:s_{2}:s_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171dc648141d444544451d5d6674e193a1b01943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.118ex; height:2.843ex;" alt="{\displaystyle (s_{1}:s_{2}:s_{3})}"> und wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\Delta =\Delta _{1}+\Delta _{2}+\Delta _{3}={\tfrac {1}{2}}(s_{1}+s_{2}+s_{3})r\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>r</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\Delta =\Delta _{1}+\Delta _{2}+\Delta _{3}={\tfrac {1}{2}}(s_{1}+s_{2}+s_{3})r\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8a21715a13806a39f683935f82f612708973c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.705ex; height:3.509ex;" alt="{\displaystyle \;\Delta =\Delta _{1}+\Delta _{2}+\Delta _{3}={\tfrac {1}{2}}(s_{1}+s_{2}+s_{3})r\;}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r={\tfrac {2\Delta }{s_{1}+s_{2}+s_{3}}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r={\tfrac {2\Delta }{s_{1}+s_{2}+s_{3}}}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af901d963355dd373233135475132d3510251b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:14.285ex; height:4.009ex;" alt="{\displaystyle \;r={\tfrac {2\Delta }{s_{1}+s_{2}+s_{3}}}\;.}"> Mit Hilfe des <a href="/wiki/Sinussatz" title="Sinussatz">Sinussatzes</a> ergibt sich für den Inkreismittelpunkt auch eine Darstellung mit den Winkeln: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(s_{1}:s_{2}:s_{3})=(\sin \varphi _{1}:\sin \varphi _{2}:\sin \varphi _{3})\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=(s_{1}:s_{2}:s_{3})=(\sin \varphi _{1}:\sin \varphi _{2}:\sin \varphi _{3})\;,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426dbb495020c2460c052425fd00ab21eafcac49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.913ex; height:2.843ex;" alt="{\displaystyle I=(s_{1}:s_{2}:s_{3})=(\sin \varphi _{1}:\sin \varphi _{2}:\sin \varphi _{3})\;,}"></dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70503774fb21be77396899900d3aa1e47d8f9e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.32ex; height:2.176ex;" alt="{\displaystyle \varphi _{i}}"> der Winkel bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> ist. Die Winkelhalbierende der Ecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}"> (Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b728e3a0b832eee6e487414f2a0cfac02e1ac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.15ex; height:2.509ex;" alt="{\displaystyle X_{3}I}">) hat die Gleichung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2}m_{1}-s_{1}m_{2}=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2}m_{1}-s_{1}m_{2}=0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951205c24cc1068f2ffa1301a8e86b358563bfa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.807ex; height:2.509ex;" alt="{\displaystyle s_{2}m_{1}-s_{1}m_{2}=0\ .}"></dd></dl> Sie schneidet die Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf212cdf059d8f70d020215c39f64f1b25c31ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}}"> (Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af7213064a4a766ce7ed1e85449eb0f04f3344d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{3}=0}">) im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{3}=(s_{1}:s_{2}:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{3}=(s_{1}:s_{2}:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd07f78f3966ea95846ee169aa9bf22d1c51389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.311ex; height:2.843ex;" alt="{\displaystyle I_{3}=(s_{1}:s_{2}:0)}">. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becba5d3350c4dd244f3cda48eb13439f21ed350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{3}}"> kann auch als Projektion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> auf die Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf212cdf059d8f70d020215c39f64f1b25c31ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}}"> angesehen werden.) Wegen (B2) gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X_{1}I_{3}|:|X_{2}I_{3}|=s_{2}:s_{1}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X_{1}I_{3}|:|X_{2}I_{3}|=s_{2}:s_{1}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401e4de079efa4af86c53687c7bce2051e83d433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.189ex; height:2.843ex;" alt="{\displaystyle |X_{1}I_{3}|:|X_{2}I_{3}|=s_{2}:s_{1}\ .}"> Analog für die anderen Winkelhalbierenden.</dd></dl> Dies ist der <a href="/wiki/Winkelhalbierendensatz_(Dreieck)" title="Winkelhalbierendensatz (Dreieck)">Winkelhalbierendensatz für das Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b385a4132a8bf83c6f72b0c857a84a205bf3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.936ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}X_{3}}"></a> . Da die Dreiecksflächen orientiert sind, kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd0ee88c6ec7be6311b5650521fa2df97196d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.736ex; height:2.509ex;" alt="{\displaystyle \Delta _{i}}"> und damit auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"> negative Werte annehmen, jenachdem, ob <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> auf derselben Seite der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> gehörigen Dreiecksseite liegt wie die Ecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> oder nicht. Beim Inkreismittelpunkt haben alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"> dasselbe Vorzeichen. Bei einem <a href="/wiki/Ankreis" title="Ankreis">Ankreismittelpunkt</a> haben (wie beim Inkreismittelpunkt) alle Abstände die Länge des Ankreisradius, aber einer der Abstände hat ein von den beiden anderen verschiedenes Vorzeichen. Damit ergeben sich die baryzentrischen Darstellungen der Ankreismittelpunkte: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=(-s_{1}:s_{2}:s_{3}),\quad A_{2}=(s_{1}:-s_{2}:s_{3}),\quad A_{3}=(s_{1}:s_{2}:-s_{3})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}=(-s_{1}:s_{2}:s_{3}),\quad A_{2}=(s_{1}:-s_{2}:s_{3}),\quad A_{3}=(s_{1}:s_{2}:-s_{3})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f949ee765126bce6d959bdc856e42f920ab4ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.406ex; height:2.843ex;" alt="{\displaystyle A_{1}=(-s_{1}:s_{2}:s_{3}),\quad A_{2}=(s_{1}:-s_{2}:s_{3}),\quad A_{3}=(s_{1}:s_{2}:-s_{3})\ .}"></dd></dl> Analog zum Inkreisradius ergibt sich für die Ankreisradien: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}={\frac {2\Delta }{-s_{1}+s_{2}+s_{3}}},\quad r_{2}={\frac {2\Delta }{s_{1}-s_{2}+s_{3}}},\quad r_{3}={\frac {2\Delta }{s_{1}+s_{2}-s_{3}}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> <mrow> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}={\frac {2\Delta }{-s_{1}+s_{2}+s_{3}}},\quad r_{2}={\frac {2\Delta }{s_{1}-s_{2}+s_{3}}},\quad r_{3}={\frac {2\Delta }{s_{1}+s_{2}-s_{3}}}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c82c3033ae220e293c28a2132dc67095faba622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:64.206ex; height:5.676ex;" alt="{\displaystyle r_{1}={\frac {2\Delta }{-s_{1}+s_{2}+s_{3}}},\quad r_{2}={\frac {2\Delta }{s_{1}-s_{2}+s_{3}}},\quad r_{3}={\frac {2\Delta }{s_{1}+s_{2}-s_{3}}}\ .}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-nagel-punkt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Baryko-nagel-punkt.svg/310px-Baryko-nagel-punkt.svg.png" decoding="async" width="310" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Baryko-nagel-punkt.svg/465px-Baryko-nagel-punkt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Baryko-nagel-punkt.svg/620px-Baryko-nagel-punkt.svg.png 2x" data-file-width="327" data-file-height="254" /></a><figcaption><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">: Nagel-Punkt. Er liegt mit dem geometrischen Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> und dem Inkreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> auf einer Gerade. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> teilt die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NI}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NI}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411266e4975d79f8378e8661f55d4fa52f494107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle NI}"> im Verhältnis 2:1</figcaption></figure> <dl><dt>Nagelpunkt</dt></dl> Aus der <a href="/wiki/Ankreis#Berührpunktabstände" title="Ankreis">Beschreibung der Lage</a> der Berührpunkte der Ankreise auf den Dreiecksseiten erkennt man ihre baryzentrische Darstellung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}=(0:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}=(0:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31db6c5545464bc96a11b95fe5b64cab563dc93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.639ex; height:2.843ex;" alt="{\displaystyle B_{1}=(0:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3}),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}=(-s_{1}+s_{2}+s_{3}:0:s_{1}+s_{2}-s_{3}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}=(-s_{1}+s_{2}+s_{3}:0:s_{1}+s_{2}-s_{3}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a2a2161f4e3741e75ce287242911ee3007f28b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.448ex; height:2.843ex;" alt="{\displaystyle B_{2}=(-s_{1}+s_{2}+s_{3}:0:s_{1}+s_{2}-s_{3}),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{3}=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:0)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{3}=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:0)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef310c36079cf256cbfb1e67841b1f14947faec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.028ex; height:2.843ex;" alt="{\displaystyle B_{3}=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:0)\ .}"></dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cda0578ec6b48774c541ecb9bee4a90176e62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle B_{i}}"> ist offensichtlich die Projektion (siehe oben) des Punktes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4973ea936a8267ca63af4a9e820b9924f708e7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.998ex; height:2.843ex;" alt="{\displaystyle N=(-s_{1}+s_{2}+s_{3}:s_{1}-s_{2}+s_{3}:s_{1}+s_{2}-s_{3})}"></dd></dl> von der Ecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> aus auf die gegenüberliegende Seite. D.h.: <dl><dd>Die drei Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X_{1}B_{1}}},{\overline {X_{2}B_{2}}},{\overline {X_{3}B_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X_{1}B_{1}}},{\overline {X_{2}B_{2}}},{\overline {X_{3}B_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c5169ccbce10112584e3a0cab6fb32eb26e96d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.803ex; height:3.343ex;" alt="{\displaystyle {\overline {X_{1}B_{1}}},{\overline {X_{2}B_{2}}},{\overline {X_{3}B_{3}}}}"> schneiden sich im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">, dem <a href="/wiki/Nagel-Punkt" title="Nagel-Punkt">Nagel-Punkt</a>.</dd></dl> Die Matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb2c084413225d30c8886870f1a7448535e64b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.951ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}}"></dd></dl> beschreibt (in baryzentrischen Koordinaten) die zentrische Streckung am geometrischen Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> mit dem Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"> (siehe Abschnitt <a href="#Steiner-Ellipse,_Steiner-Inellipse">Steiner-Ellipse, Steiner-Inellipse</a>). Bildet man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> damit ab, erhält man den Inkreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}">. Dies zeigt: <dl><dd>Die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,S,I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N,S,I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2886fbe11a2127d21f1f96a7198627959914a9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.802ex; height:2.509ex;" alt="{\displaystyle N,S,I}"> liegen auf einer Gerade durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> teilt die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NI}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NI}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411266e4975d79f8378e8661f55d4fa52f494107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle NI}"> im Verhältnis 2:1.</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-umkreismittelpunkt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Baryko-umkreismittelpunkt.svg/220px-Baryko-umkreismittelpunkt.svg.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Baryko-umkreismittelpunkt.svg/330px-Baryko-umkreismittelpunkt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Baryko-umkreismittelpunkt.svg/440px-Baryko-umkreismittelpunkt.svg.png 2x" data-file-width="221" data-file-height="207" /></a><figcaption>Umkreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></figcaption></figure> <dl><dt>Umkreismittelpunkt</dt></dl> Der Umkreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> hat zu den Ecken den gleichen Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, den Umkreisradius. Der Winkel bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> im Teildreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be92ab4984621bd94d8e4683686dfc14ef598d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.807ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},U}"> ist wegen des <a href="/wiki/Kreiswinkelsatz" class="mw-redirect" title="Kreiswinkelsatz">Kreiswinkelsatzes</a> doppelt so groß wie der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6b64fd8e7ea9f0b659c8d18e9d1e61420adc50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{3}}"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}">. Also ist die <a href="/wiki/Dreiecksfl%C3%A4che" title="Dreiecksfläche">Fläche</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{3}={\tfrac {1}{2}}R^{2}\sin 2\varphi _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{3}={\tfrac {1}{2}}R^{2}\sin 2\varphi _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06468f67b0825869c8e8fe8408381034ebea987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.932ex; height:3.509ex;" alt="{\displaystyle \Delta _{3}={\tfrac {1}{2}}R^{2}\sin 2\varphi _{3}}">. Entsprechendes gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1},\Delta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1},\Delta _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f877ca0245cb2561e8e3b15c9584f06f0cb58fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.014ex; height:2.509ex;" alt="{\displaystyle \Delta _{1},\Delta _{2}}">. Damit sind die baryzentrischen Koordinaten des Umkreismittelpunktes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sin 2\varphi _{1}:\sin 2\varphi _{2}:\sin 2\varphi _{3})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sin 2\varphi _{1}:\sin 2\varphi _{2}:\sin 2\varphi _{3})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7c458ebd4c1f7aee9a98b15188d9df16a89bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.85ex; height:2.843ex;" alt="{\displaystyle (\sin 2\varphi _{1}:\sin 2\varphi _{2}:\sin 2\varphi _{3})\ .}"></dd></dl> Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sin 2\varphi _{i}=2\sin \varphi _{i}\cos \varphi _{i},\;\sin \varphi _{i}={\tfrac {s_{i}}{2R}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sin 2\varphi _{i}=2\sin \varphi _{i}\cos \varphi _{i},\;\sin \varphi _{i}={\tfrac {s_{i}}{2R}}\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a723cfde5b97ba8cd92b7c89bf1f1f4360d3eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:37.677ex; height:3.676ex;" alt="{\displaystyle \;\sin 2\varphi _{i}=2\sin \varphi _{i}\cos \varphi _{i},\;\sin \varphi _{i}={\tfrac {s_{i}}{2R}}\;}"> und den <a href="/wiki/Kosinussatz" title="Kosinussatz">Kosinussätzen</a> für die drei Winkel ergibt sich die winkelfreie Darstellung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}s_{1}^{2}(-s_{1}^{2}+s_{2}^{2}+s_{3}^{2}):s_{2}^{2}(s_{1}^{2}-s_{2}^{2}+s_{3}^{2}):s_{3}^{2}(s_{1}^{2}+s_{2}^{2}-s_{3}^{2}){\big )}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>:</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>:</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}s_{1}^{2}(-s_{1}^{2}+s_{2}^{2}+s_{3}^{2}):s_{2}^{2}(s_{1}^{2}-s_{2}^{2}+s_{3}^{2}):s_{3}^{2}(s_{1}^{2}+s_{2}^{2}-s_{3}^{2}){\big )}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61abd9d39bb2fcb36ebb69b7d0a944fe6187cf68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.246ex; height:3.176ex;" alt="{\displaystyle {\big (}s_{1}^{2}(-s_{1}^{2}+s_{2}^{2}+s_{3}^{2}):s_{2}^{2}(s_{1}^{2}-s_{2}^{2}+s_{3}^{2}):s_{3}^{2}(s_{1}^{2}+s_{2}^{2}-s_{3}^{2}){\big )}\ .}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-hoehenschnittpunkt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Baryko-hoehenschnittpunkt.svg/220px-Baryko-hoehenschnittpunkt.svg.png" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Baryko-hoehenschnittpunkt.svg/330px-Baryko-hoehenschnittpunkt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Baryko-hoehenschnittpunkt.svg/440px-Baryko-hoehenschnittpunkt.svg.png 2x" data-file-width="218" data-file-height="201" /></a><figcaption>Höhenschnittpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></figcaption></figure> <dl><dt>Höhenschnittpunkt</dt></dl> Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=(m_{1}:m_{2}:m_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=(m_{1}:m_{2}:m_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5413bed2bd36b5639a39fb535c2991952ba45d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.13ex; height:2.843ex;" alt="{\displaystyle H=(m_{1}:m_{2}:m_{3})}"> der Höhenschnittpunkt, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}=(m_{1}:m_{2}:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}=(m_{1}:m_{2}:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d7363e20fe969119abb30a2a0ad11a089362de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.68ex; height:2.843ex;" alt="{\displaystyle P_{3}=(m_{1}:m_{2}:0)}"> der Fußpunkt der Höhe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b7b4e24b0477e3270e7c640a73be1a8dc7f300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{3}}"> (siehe Bild) und es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\tan \varphi _{1}={\tfrac {h_{3}}{|P_{3}X_{1}|}},\;\tan \varphi _{2}={\tfrac {h_{3}}{|P_{3}X_{2}|}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\tan \varphi _{1}={\tfrac {h_{3}}{|P_{3}X_{1}|}},\;\tan \varphi _{2}={\tfrac {h_{3}}{|P_{3}X_{2}|}}\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbadf8bb0d817b361783395888edfbd2c933169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.468ex; height:4.676ex;" alt="{\displaystyle \;\tan \varphi _{1}={\tfrac {h_{3}}{|P_{3}X_{1}|}},\;\tan \varphi _{2}={\tfrac {h_{3}}{|P_{3}X_{2}|}}\;}"> Wegen (B2) ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\tan \varphi _{1}:\tan \varphi _{2}=|P_{3}X_{2}|:|P_{3}X_{1}|=m_{1}:m_{2}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\tan \varphi _{1}:\tan \varphi _{2}=|P_{3}X_{2}|:|P_{3}X_{1}|=m_{1}:m_{2}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f84196c9ecba5af03051aa03f04093402017375" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.414ex; height:2.843ex;" alt="{\displaystyle \;\tan \varphi _{1}:\tan \varphi _{2}=|P_{3}X_{2}|:|P_{3}X_{1}|=m_{1}:m_{2}\;.}"> Analog ergeben sich die anderen Verhältnisse. Damit hat der Höhenschnittpunkt die baryzentrischen Koordinaten <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tan \varphi _{1}:\tan \varphi _{2}:\tan \varphi _{3})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\tan \varphi _{1}:\tan \varphi _{2}:\tan \varphi _{3})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de54cc5341c602bc62eaa81b58663310a84ea472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.874ex; height:2.843ex;" alt="{\displaystyle (\tan \varphi _{1}:\tan \varphi _{2}:\tan \varphi _{3})\ .}"></dd></dl> Falls ein Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"> ist, z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{3}=90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{3}=90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f3df0da503c93e9b4da7bd7db5fa45071156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.052ex; height:2.843ex;" alt="{\displaystyle \varphi _{3}=90^{\circ }}">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c954b74e301d76dae12d5be7bb31e20aecf98e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.141ex; height:2.509ex;" alt="{\displaystyle H=X_{3}}">. <dl><dt>Spieker-Punkt</dt></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-3eck-Spieker-P.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Baryko-3eck-Spieker-P.svg/290px-Baryko-3eck-Spieker-P.svg.png" decoding="async" width="290" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Baryko-3eck-Spieker-P.svg/435px-Baryko-3eck-Spieker-P.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Baryko-3eck-Spieker-P.svg/580px-Baryko-3eck-Spieker-P.svg.png 2x" data-file-width="255" data-file-height="163" /></a><figcaption>Spieker-Punkt eines Dreiecks</figcaption></figure> Belegt man die Seiten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}X_{3},X_{3}X_{1},X_{1}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}X_{3},X_{3}X_{1},X_{1}X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995e1ef1455446a48d5bc27e713bd58af3f0fc5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.939ex; height:2.509ex;" alt="{\displaystyle X_{2}X_{3},X_{3}X_{1},X_{1}X_{2}}"> eines Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b385a4132a8bf83c6f72b0c857a84a205bf3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.936ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}X_{3}}"> gleichmäßig mit Masse, so nennt man den zugehörigen Kantenschwerpunkt <a href="/wiki/Spieker-Punkt" title="Spieker-Punkt">Spieker-Punkt</a>. (Ecken- und Flächenschwerpunkt eines Dreiecks <a href="/wiki/Geometrischer_Schwerpunkt#Dreieck" title="Geometrischer Schwerpunkt">sind identisch</a>: der Schnittpunkt der Seitenhalbierenden.) Denkt man sich die Masse einer Seite in ihrem Schwerpunkt, dem Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"> konzentriert, so ist der Spieker-Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}=(x_{s},y_{s})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=(x_{s},y_{s})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bff2bb1f716282ba328ab0f90ac71bf26fd6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.91ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}=(x_{s},y_{s})}"> der Schwerpunkt des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}M_{2}M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}M_{2}M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35768d7669dc9189b85749b2662f0d61add8399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.925ex; height:2.509ex;" alt="{\displaystyle M_{1}M_{2}M_{3}}"> mit den Seitenlängen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},s_{2},s_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1},s_{2},s_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3e829da5f828c27f32835462b5bdf754ab1c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.502ex; height:2.009ex;" alt="{\displaystyle s_{1},s_{2},s_{3}}"> als Massenbelegungen in den Ecken. Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=({\tfrac {x_{2}+x_{3}}{2}},{\tfrac {y_{2}+y_{3}}{2}}),\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}=({\tfrac {x_{2}+x_{3}}{2}},{\tfrac {y_{2}+y_{3}}{2}}),\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724dbb58c7ee94b754aad0a27531453de08ad7c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.054ex; height:3.843ex;" alt="{\displaystyle M_{1}=({\tfrac {x_{2}+x_{3}}{2}},{\tfrac {y_{2}+y_{3}}{2}}),\dots }"> und (S3) folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{s}={\frac {s_{1}{\frac {x_{2}+x_{3}}{2}}+s_{2}{\frac {x_{1}+x_{3}}{2}}+s_{3}{\frac {x_{1}+x_{2}}{2}}}{s_{1}+s_{2}+s_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{s}={\frac {s_{1}{\frac {x_{2}+x_{3}}{2}}+s_{2}{\frac {x_{1}+x_{3}}{2}}+s_{3}{\frac {x_{1}+x_{2}}{2}}}{s_{1}+s_{2}+s_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b49f06ac315df8992ae318cbcd3ad077d03a6be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.357ex; height:7.009ex;" alt="{\displaystyle x_{s}={\frac {s_{1}{\frac {x_{2}+x_{3}}{2}}+s_{2}{\frac {x_{1}+x_{3}}{2}}+s_{3}{\frac {x_{1}+x_{2}}{2}}}{s_{1}+s_{2}+s_{3}}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad ={\frac {(s_{2}+s_{3})x_{1}+(s_{1}+s_{3})x_{2}+(s_{1}+s_{2})x_{3}}{2(s_{1}+s_{2}+s_{3})}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad ={\frac {(s_{2}+s_{3})x_{1}+(s_{1}+s_{3})x_{2}+(s_{1}+s_{2})x_{3}}{2(s_{1}+s_{2}+s_{3})}}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f6c36b85c14297aba3151b0f5fd4a4600d611f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.489ex; height:6.509ex;" alt="{\displaystyle \quad ={\frac {(s_{2}+s_{3})x_{1}+(s_{1}+s_{3})x_{2}+(s_{1}+s_{2})x_{3}}{2(s_{1}+s_{2}+s_{3})}}\ .}"></dd></dl> Analog ergibt sich die y-Koordinate. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-3eck-Spieker-inkr.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Baryko-3eck-Spieker-inkr.svg/290px-Baryko-3eck-Spieker-inkr.svg.png" decoding="async" width="290" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Baryko-3eck-Spieker-inkr.svg/435px-Baryko-3eck-Spieker-inkr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Baryko-3eck-Spieker-inkr.svg/580px-Baryko-3eck-Spieker-inkr.svg.png 2x" data-file-width="255" data-file-height="163" /></a><figcaption>Spieker-Punkt als Mittelpunkt des Inkreises des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}M_{2}M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}M_{2}M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35768d7669dc9189b85749b2662f0d61add8399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.925ex; height:2.509ex;" alt="{\displaystyle M_{1}M_{2}M_{3}}"></figcaption></figure> Hieraus erkennt man die baryzentrischen Koordinaten des Spieker-Punktes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}=(s_{2}+s_{3}:s_{1}+s_{3}:s_{1}+s_{2})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=(s_{2}+s_{3}:s_{1}+s_{3}:s_{1}+s_{2})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8822f4dd0bfa4f1620ef78c35e88f1305621f3bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.891ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}=(s_{2}+s_{3}:s_{1}+s_{3}:s_{1}+s_{2})\ .}"></dd></dl> Bedeutung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}"> für das Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}M_{2}M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}M_{2}M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35768d7669dc9189b85749b2662f0d61add8399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.925ex; height:2.509ex;" alt="{\displaystyle M_{1}M_{2}M_{3}}">: Aus den obigen Überlegungen (Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}">) folgt direkt die baryzentrische Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}"> bezüglich des (grünen) Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}M_{2}M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}M_{2}M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35768d7669dc9189b85749b2662f0d61add8399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.925ex; height:2.509ex;" alt="{\displaystyle M_{1}M_{2}M_{3}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}=(s_{1}:s_{2}:s_{3})_{M}={\big (}{\tfrac {s_{1}}{2}}:{\tfrac {s_{2}}{2}}:{\tfrac {s_{3}}{2}}{\big )}_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=(s_{1}:s_{2}:s_{3})_{M}={\big (}{\tfrac {s_{1}}{2}}:{\tfrac {s_{2}}{2}}:{\tfrac {s_{3}}{2}}{\big )}_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec66a0a9484dfbd5f5e499966e140d736766aeaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.046ex; height:3.509ex;" alt="{\displaystyle {\mathcal {S}}=(s_{1}:s_{2}:s_{3})_{M}={\big (}{\tfrac {s_{1}}{2}}:{\tfrac {s_{2}}{2}}:{\tfrac {s_{3}}{2}}{\big )}_{M}}"></dd></dl> Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {s_{i}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {s_{i}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b40dac07241b0b99b3ad2212bd6ad2601053b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.232ex; height:3.509ex;" alt="{\displaystyle {\tfrac {s_{i}}{2}}}"> die Länge der dem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"> gegenüberliegenden (grünen) Seite ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}"> der Inkreismittelpunkt = Schnittpunkt der Winkelhalbierenden des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}M_{2}M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}M_{2}M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35768d7669dc9189b85749b2662f0d61add8399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.925ex; height:2.509ex;" alt="{\displaystyle M_{1}M_{2}M_{3}}"> (siehe oben). Diese Eigenschaft liefert die Möglichkeit den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}"> zeichnerisch zu bestimmen. <dl><dt>Eulergerade</dt></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-euler-gerade.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Baryko-euler-gerade.svg/290px-Baryko-euler-gerade.svg.png" decoding="async" width="290" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Baryko-euler-gerade.svg/435px-Baryko-euler-gerade.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Baryko-euler-gerade.svg/580px-Baryko-euler-gerade.svg.png 2x" data-file-width="292" data-file-height="282" /></a><figcaption>Eulergerade eines Dreiecks</figcaption></figure> Der geometrische Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, der Umkreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> und der Höhenschnittpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> liegen auf einer Gerade, der <a href="/wiki/Eulergerade" class="mw-redirect" title="Eulergerade">Eulergerade</a>. Denn, führt man am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> eine <a href="/wiki/Zentrische_Streckung" title="Zentrische Streckung">zentrische Streckung</a> mit Streckfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"> durch, wird jede Ecke auf den Mittelpunkt der ihr gegenüberliegenden Seite abgebildet (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> teilt jede Seitenhalbierende im Verhältnis 2:1) und die Höhen werden auf die Mittelsenkrechten abgebildet. Also geht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> über und beide Punkte liegen auf einer gemeinsamen Gerade durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">. Der Umkreis geht dabei in den Kreis durch die Seitenmitten, den <a href="/wiki/Feuerbachkreis" title="Feuerbachkreis">Feuerbachkreis</a>, über, dessen Mittelpunkt (Bild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">) also auch auf der Eulergerade liegt. Die Gleichung der Eulergerade in baryzentrischen Koordinaten ist (s. oben) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\1&1&1\\\sin 2\varphi _{1}&\sin 2\varphi _{2}&\sin 2\varphi _{3}\end{matrix}}\right|=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\1&1&1\\\sin 2\varphi _{1}&\sin 2\varphi _{2}&\sin 2\varphi _{3}\end{matrix}}\right|=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3520cb139c786da01aa7a2c46e0d87baccc78f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.05ex; height:9.509ex;" alt="{\displaystyle \ \left|{\begin{matrix}m_{1}&m_{2}&m_{3}\\1&1&1\\\sin 2\varphi _{1}&\sin 2\varphi _{2}&\sin 2\varphi _{3}\end{matrix}}\right|=}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ m_{1}(\sin 2\varphi _{3}-\sin 2\varphi _{2})+m_{2}(\sin 2\varphi _{1}-\sin 2\varphi _{3})+m_{3}(\sin 2\varphi _{2}-\sin 2\varphi _{1})=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ m_{1}(\sin 2\varphi _{3}-\sin 2\varphi _{2})+m_{2}(\sin 2\varphi _{1}-\sin 2\varphi _{3})+m_{3}(\sin 2\varphi _{2}-\sin 2\varphi _{1})=0\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b4b958272ee3b1b5bf297776ccdfe12b3863f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.213ex; height:2.843ex;" alt="{\displaystyle \ m_{1}(\sin 2\varphi _{3}-\sin 2\varphi _{2})+m_{2}(\sin 2\varphi _{1}-\sin 2\varphi _{3})+m_{3}(\sin 2\varphi _{2}-\sin 2\varphi _{1})=0\ }"></dd></dl> oder unter Verwendung von Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ m_{1}(\tan \varphi _{3}-\tan \varphi _{2})+m_{2}(\tan \varphi _{1}-\tan \varphi _{3})+m_{3}(\tan \varphi _{2}-\tan \varphi _{1})=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ m_{1}(\tan \varphi _{3}-\tan \varphi _{2})+m_{2}(\tan \varphi _{1}-\tan \varphi _{3})+m_{3}(\tan \varphi _{2}-\tan \varphi _{1})=0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b3d5b55b7d5e308d2d723423c91fa19350ec69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.909ex; height:2.843ex;" alt="{\displaystyle \ m_{1}(\tan \varphi _{3}-\tan \varphi _{2})+m_{2}(\tan \varphi _{1}-\tan \varphi _{3})+m_{3}(\tan \varphi _{2}-\tan \varphi _{1})=0\ .}"></dd></dl> Gleichseitige Dreiecke besitzen keine Eulergerade, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=H=U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>H</mi> <mo>=</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=H=U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bece7a5addd3df4f9e97ccb447423298dccacd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.542ex; height:2.176ex;" alt="{\displaystyle S=H=U}"> ist. Ist das Dreieck gleichschenklig, aber nicht gleichseitig, z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}=\varphi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{1}=\varphi _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2644c145bd19634cf3e99575671683c1a6d399f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.247ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}=\varphi _{2}}">, so hat die Eulergerade die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;m_{1}-m_{2}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m_{1}-m_{2}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045a250160be271259d0e74f202db0ac84b1976c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.581ex; height:2.509ex;" alt="{\displaystyle \;m_{1}-m_{2}=0\;}"> und ist gleich der Seitenhalbierenden durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}">. Sie enthält dann auch den Inkreismittelpunkt. Ist das Dreieck rechtwinklig, z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{3}=90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{3}=90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f3df0da503c93e9b4da7bd7db5fa45071156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.052ex; height:2.843ex;" alt="{\displaystyle \varphi _{3}=90^{\circ }}">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}=90^{\circ }-\varphi _{1}\;\to \;\sin 2\varphi _{2}=\sin 2\varphi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{2}=90^{\circ }-\varphi _{1}\;\to \;\sin 2\varphi _{2}=\sin 2\varphi _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aecb2481deaa12af459c681c634174cd221a02b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.429ex; height:2.843ex;" alt="{\displaystyle \varphi _{2}=90^{\circ }-\varphi _{1}\;\to \;\sin 2\varphi _{2}=\sin 2\varphi _{1}}"> und die Eulergerade hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;m_{1}-m_{2}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m_{1}-m_{2}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045a250160be271259d0e74f202db0ac84b1976c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.581ex; height:2.509ex;" alt="{\displaystyle \;m_{1}-m_{2}=0\;}"> und ist die Seitenhalbierende zur Hypotenuse. <div class="mw-heading mw-heading3"><h3 id="Satz_von_Ceva">Satz von Ceva</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=18" title="Abschnitt bearbeiten: Satz von Ceva" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Satz von Ceva">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-ceva.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Baryzentr-ko-ceva.svg/310px-Baryzentr-ko-ceva.svg.png" decoding="async" width="310" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Baryzentr-ko-ceva.svg/465px-Baryzentr-ko-ceva.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Baryzentr-ko-ceva.svg/620px-Baryzentr-ko-ceva.svg.png 2x" data-file-width="256" data-file-height="195" /></a><figcaption>Satz von Ceva</figcaption></figure> <dl><dt><a href="/wiki/Satz_von_Ceva" title="Satz von Ceva">Satz von Ceva</a></dt></dl> Ist P ein Punkt innerhalb des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4126aa4015fd02d5478fd2d09601afd33286509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> der Schnittpunkt der Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {PX_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {PX_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440407d0dd4f2237cfdc8361d9adc41b9154bf10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.584ex; height:3.343ex;" alt="{\displaystyle {\overline {PX_{i}}}}"> mit der Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}X_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c82d08dad6ed1bb058de52f29b82f56bffe142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.847ex; height:2.843ex;" alt="{\displaystyle X_{j}X_{k}}"> (siehe Bild), so gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}=1\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}=1\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb679b2776aa5fd868eb0285323e24e661e7ca64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.876ex; height:6.509ex;" alt="{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}=1\;.}"></dd></dl> <dl><dt>Beweis</dt></dl> Mit den Punkten in baryzentrischen Koordinaten: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1:0:0),\;X_{2}=(0:1:0),\;X_{3}=(0:0:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1:0:0),\;X_{2}=(0:1:0),\;X_{3}=(0:0:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14099d6739178db5fcd212a3aa23d4fda6dd6881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.102ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1:0:0),\;X_{2}=(0:1:0),\;X_{3}=(0:0:1)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(m_{1}:m_{2}:m_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(m_{1}:m_{2}:m_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3e04d6c0dab763f983db2fb33728c3831137a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.811ex; height:2.843ex;" alt="{\displaystyle P=(m_{1}:m_{2}:m_{3})}"></dd></dl> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=(0:m_{2}:m_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=(0:m_{2}:m_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4fe1a23b37f1bf82bc5fdc96f5093651607cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.68ex; height:2.843ex;" alt="{\displaystyle P_{1}=(0:m_{2}:m_{3})}"> (siehe Besondere Punkte). Aus B2 erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X_{2}P_{1}|:|X_{3}P_{1}|=m_{3}:m_{2}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X_{2}P_{1}|:|X_{3}P_{1}|=m_{3}:m_{2}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fededa1d95f2edeb8b643fea44436ec106c0fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.091ex; height:2.843ex;" alt="{\displaystyle |X_{2}P_{1}|:|X_{3}P_{1}|=m_{3}:m_{2}\;.}"> Führt man diese Überlegungen auch für die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2},P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2},P_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898fae629cca84b7757189661a867ae1c240cec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.127ex; height:2.509ex;" alt="{\displaystyle P_{2},P_{3}}"> durch, ergibt sich <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}={\frac {m_{3}}{m_{2}}}\cdot {\frac {m_{1}}{m_{3}}}\cdot {\frac {m_{2}}{m_{1}}}=1\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}={\frac {m_{3}}{m_{2}}}\cdot {\frac {m_{1}}{m_{3}}}\cdot {\frac {m_{2}}{m_{1}}}=1\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dc917e13fcab536ff578c78245eb0f224e921c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.125ex; height:6.509ex;" alt="{\displaystyle {\frac {|X_{2}P_{1}|}{|X_{3}P_{1}|}}\cdot {\frac {|X_{3}P_{2}|}{|X_{1}P_{2}|}}\cdot {\frac {|X_{1}P_{3}|}{|X_{2}P_{3}|}}={\frac {m_{3}}{m_{2}}}\cdot {\frac {m_{1}}{m_{3}}}\cdot {\frac {m_{2}}{m_{1}}}=1\;.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Steiner-Ellipse,_Steiner-Inellipse">Steiner-Ellipse, Steiner-Inellipse</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=19" title="Abschnitt bearbeiten: Steiner-Ellipse, Steiner-Inellipse" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Steiner-Ellipse, Steiner-Inellipse">Quelltext bearbeiten</a>]</div> Die eindeutig bestimmte Ellipse durch die Ecken des (beliebigen) Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4126aa4015fd02d5478fd2d09601afd33286509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}}">, deren Mittelpunkt der geometrische Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> ist, heißt <a href="/wiki/Steiner-Ellipse" title="Steiner-Ellipse">Steiner-Ellipse</a>. In baryzentrischen Koordinaten wird sie durch die Gleichung <dl><dd>(SE)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82804019479508d9b36daa577c7798f1ac9b3a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.832ex; height:2.509ex;" alt="{\displaystyle \quad m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0}"></dd></dl> beschrieben. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-steiner-ellipse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Baryko-steiner-ellipse.svg/220px-Baryko-steiner-ellipse.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Baryko-steiner-ellipse.svg/330px-Baryko-steiner-ellipse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Baryko-steiner-ellipse.svg/440px-Baryko-steiner-ellipse.svg.png 2x" data-file-width="219" data-file-height="221" /></a><figcaption>Steiner-Ellipse</figcaption></figure> Man prüft leicht nach, dass die sechs Punkte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1:0:0),\ X_{2}=(0:1:0),\ \ X_{3}=(0:0:1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1:0:0),\ X_{2}=(0:1:0),\ \ X_{3}=(0:0:1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ded5bdd88d3f4c60ba7fe1041870dd83635a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.201ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1:0:0),\ X_{2}=(0:1:0),\ \ X_{3}=(0:0:1),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1}=(-1:2:2),Y_{2}=(2:-1:2),Y_{3}=(2:2:-1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>2</mn> <mo>:</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>:</mo> <mn>2</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{1}=(-1:2:2),Y_{2}=(2:-1:2),Y_{3}=(2:2:-1)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f9c91c0323fb9ffef418b4ba9e00dcde449858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.16ex; height:2.843ex;" alt="{\displaystyle Y_{1}=(-1:2:2),Y_{2}=(2:-1:2),Y_{3}=(2:2:-1)\;}"></dd></dl> die Gleichung (SE) erfüllen und, dass der Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(1:1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(1:1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e82b5828f9618ee60c465ad6aa07e62b7bb96cd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.769ex; height:2.843ex;" alt="{\displaystyle S=(1:1:1)}"> der Mittelpunkt (siehe Abschnitt Definition) der Paare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i},Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i},Y_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf882cf6f13deb2bbcaf1aee52a955c50d9c869d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.908ex; height:2.509ex;" alt="{\displaystyle X_{i},Y_{i}}"> ist. Die Gleichung (SE) muss also einen nicht ausgearteten Kegelschnitt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">k</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2150bf2c3d77f7e44c9408400df3e22f24912f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle {\mathcal {k}}}"> (Ellipse oder Hyperbel oder Parabel) beschreiben. Da aus den Gleichungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0,\quad m_{1}+m_{2}+m_{3}=0\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0,\quad m_{1}+m_{2}+m_{3}=0\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf5c2080963c58f4839d466223f3d74eca5dd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.414ex; height:2.509ex;" alt="{\displaystyle m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1}=0,\quad m_{1}+m_{2}+m_{3}=0\quad }"> der Widerspruch</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=(m_{1}+m_{2}+m_{3})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=(m_{1}+m_{2}+m_{3})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974341c99e184babb8f7fa2a3cd7c1eb99adfeee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.089ex; height:3.176ex;" alt="{\displaystyle 0=(m_{1}+m_{2}+m_{3})^{2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a576bf4b79d8b3a784b4101ae3ae6aa9074a6e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.705ex; height:3.176ex;" alt="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\neq 0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>≠</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\neq 0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e6a004bf929c6826fbe9f667640447cf8097de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.132ex; height:3.176ex;" alt="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\neq 0\ .}"></dd></dl> folgt, hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">k</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2150bf2c3d77f7e44c9408400df3e22f24912f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle {\mathcal {k}}}"> mit der Ferngerade keinen Punkt gemeinsam, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">k</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2150bf2c3d77f7e44c9408400df3e22f24912f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle {\mathcal {k}}}"> ist eine Ellipse. Die Spiegelung am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> lässt das Sechseck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}X_{3}Y_{1}Y_{2}Y_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}X_{3}Y_{1}Y_{2}Y_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6df087e97f7fb351e2b88d9a6dc45130c1848f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.15ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}X_{3}Y_{1}Y_{2}Y_{3}}"> und damit auch die Ellipse invariant (Eine Ellipse ist durch 5 ihrer Punkte eindeutig bestimmt). Also ist der Symmetriepunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> der Mittelpunkt der Ellipse. Da der Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba9967ecd86d57a9d85001e5fe15f93686861d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}"> der Sehne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf212cdf059d8f70d020215c39f64f1b25c31ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}}"> auf dem Durchmesser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}Y_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}Y_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5fe4ad6e4049540605867b9d079c4569da1f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.383ex; height:2.509ex;" alt="{\displaystyle X_{3}Y_{3}}"> liegt, muss die Tangente in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}"> parallel zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf212cdf059d8f70d020215c39f64f1b25c31ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{2}}"> sein (siehe <a href="/wiki/Ellipse#Konjugierte_Durchmesser" title="Ellipse">Ellipse</a>). Sie hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83d9b7fca4fa06e928961412c2ac9176ae8d33b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.291ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}=0}">. Schneidet man die Parallele zur Tangente durch den Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> (sie hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}-2m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}-2m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd319cb43cab115ffba1078f1d86c6b8fdaf6e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.388ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}-2m_{3}=0}">) mit der Ellipse (SE) erhält man die zwei zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}"> konjugierten Punkte (siehe <a href="/wiki/Steiner-Ellipse#Zeichnerische_Bestimmung_der_Steiner-Ellipse" title="Steiner-Ellipse">Steiner-Ellipse</a>) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3}=(1-{\sqrt {3}}:1+{\sqrt {3}}:1),\quad D'_{3}=(1+{\sqrt {3}}:1-{\sqrt {3}}:1)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>:</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>:</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{3}=(1-{\sqrt {3}}:1+{\sqrt {3}}:1),\quad D'_{3}=(1+{\sqrt {3}}:1-{\sqrt {3}}:1)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb02aa4a7ffbfaf20221f945fb9f9252c208dd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.835ex; height:3.176ex;" alt="{\displaystyle D_{3}=(1-{\sqrt {3}}:1+{\sqrt {3}}:1),\quad D'_{3}=(1+{\sqrt {3}}:1-{\sqrt {3}}:1)\ .}"></dd></dl> Entsprechendes gilt für die Tangenten in den anderen Ecken. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-steiner-inellipse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Baryko-steiner-inellipse.svg/220px-Baryko-steiner-inellipse.svg.png" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Baryko-steiner-inellipse.svg/330px-Baryko-steiner-inellipse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Baryko-steiner-inellipse.svg/440px-Baryko-steiner-inellipse.svg.png 2x" data-file-width="224" data-file-height="244" /></a><figcaption>Steiner-Inellipse (grün)</figcaption></figure> Bildet man die Steiner-Ellipse mit der zentrischen Streckung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> an ihrem Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> mit Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"> ab, erhält man also eine Ellipse mit demselben Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, die die Dreiecksseiten in deren Mittelpunkten berührt. Dies ist die <a href="/wiki/Steiner-Inellipse" title="Steiner-Inellipse">Steiner-Inellipse</a> des Dreiecks. Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;M_{1}=(0:1:1),\;M_{2}=(1:0:1),\;M_{3}=(1:1:0)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;M_{1}=(0:1:1),\;M_{2}=(1:0:1),\;M_{3}=(1:1:0)\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a1e8072cadf9c29d058445e9f150b90aa190bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.382ex; height:2.843ex;" alt="{\displaystyle \;M_{1}=(0:1:1),\;M_{2}=(1:0:1),\;M_{3}=(1:1:0)\;}"> ist die Abbildungsmatrix von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a85e82990df4348cc2f8e03bfa532872586e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.178ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}\ .}"></dd></dl> Transformiert man die Gleichung (SE) der Steiner-Ellipse mit dieser Matrix, ergibt sich die Gleichung der Steiner-Inellipse in baryzentrischen Koordinaten: <dl><dd>(SIE)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})=0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9954b2f3175bd63c6f4a7df1990ca7f4c11dbbad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.836ex; height:3.176ex;" alt="{\displaystyle \quad m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})=0\ .}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-steiner-ellipsen-3d.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Baryko-steiner-ellipsen-3d.svg/220px-Baryko-steiner-ellipsen-3d.svg.png" decoding="async" width="220" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Baryko-steiner-ellipsen-3d.svg/330px-Baryko-steiner-ellipsen-3d.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Baryko-steiner-ellipsen-3d.svg/440px-Baryko-steiner-ellipsen-3d.svg.png 2x" data-file-width="225" data-file-height="294" /></a><figcaption>Steiner-Ellipsen als Kegel in (homogenen) baryzentrischen Koordinaten und in normierten baryzentrischen Koordinaten als Kreise in der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa64a502bfee18a11f4430e03682542f1cc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.226ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> </figcaption></figure> <dl><dt>3d-Darstellungen</dt></dl> 1) Die durch die Gleichung (SE) definierte Quadrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Q}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Q}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b788c6cdc4ae9541835467f094663527492acbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.953ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Q}}_{1}}"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> mit (wie üblich) orthogonalen Koordinatenachsen ist ein gerader <a href="/wiki/Kreiskegel" class="mw-redirect" title="Kreiskegel">Kreiskegel</a> mit dem Nullpunkt als Spitze, der die Koordinatenachsen enthält und die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(1,1,1)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(1,1,1)^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86698f280501057f6f71068f0cf4a053fe16e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.593ex; height:3.176ex;" alt="{\displaystyle t(1,1,1)^{T}}"> als Achse besitzt. Denn für die Schnittkurve der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa64a502bfee18a11f4430e03682542f1cc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.226ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> und der Quadrik mit der Gleichung (SE) gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=(m_{1}+m_{2}+m_{3})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=(m_{1}+m_{2}+m_{3})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea842c3e1d1391154dee90171293bcaabbd10013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.089ex; height:3.176ex;" alt="{\displaystyle 1=(m_{1}+m_{2}+m_{3})^{2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a576bf4b79d8b3a784b4101ae3ae6aa9074a6e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.705ex; height:3.176ex;" alt="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ac9266740851deaf9146e4031e2773fdc6d5c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.871ex; height:3.176ex;" alt="{\displaystyle \ =m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\ .}"></dd></dl> D.h.: die Schnittkurve ist auch ein ebener Schnitt der Einheitskugel und damit ein Kreis (im Bild lila). 2) Analoge Überlegungen für die durch die Gleichung (SIE) definierte Quadrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Q}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Q}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baecaafc7e7b85ec8701403cf84b5bbf99c94bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.953ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Q}}_{2}}"> zeigen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Q}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Q}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baecaafc7e7b85ec8701403cf84b5bbf99c94bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.953ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Q}}_{2}}"> ist auch ein gerader Kreiskegel mit dem Nullpunkt als Spitze und der Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(1,1,1)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(1,1,1)^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86698f280501057f6f71068f0cf4a053fe16e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.593ex; height:3.176ex;" alt="{\displaystyle t(1,1,1)^{T}}"> als Achse. Der Basiskreis ist der Schnitt der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa64a502bfee18a11f4430e03682542f1cc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.226ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> mit der kleineren Kugel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}^{2}+m_{2}^{2}+m_{3}^{2}={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}^{2}+m_{2}^{2}+m_{3}^{2}={\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2366e2a2efbd0341a72ace7a6444fcd68517fae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.721ex; height:3.509ex;" alt="{\displaystyle m_{1}^{2}+m_{2}^{2}+m_{3}^{2}={\tfrac {1}{2}}}"> (im Bild grün). Schneidet man den Kegel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Q}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Q}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baecaafc7e7b85ec8701403cf84b5bbf99c94bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.953ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Q}}_{2}}"> mit der Koordinatenebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811e411751820532576f97517df840acb30c4dbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{1}=0}">, ergibt sich die Ursprungsgerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(0,1,1)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(0,1,1)^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b63ff9406854b401d856be609147910b015278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.593ex; height:3.176ex;" alt="{\displaystyle t(0,1,1)^{T}}">, d. h. der Kegel berührt die Koordinatenebene. Dies gilt auch für die anderen Koordinatenebenen. 3) In normierten baryzentrischen Koordinaten (d. h. in der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}+m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}+m_{2}+m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa64a502bfee18a11f4430e03682542f1cc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.226ex; height:2.509ex;" alt="{\displaystyle m_{1}+m_{2}+m_{3}=1}">) erscheint das gegebene Dreieck gleichseitig und die Steiner-Ellipsen sind dessen Umkreis und Inkreis. 4) Setzt man keine orthogonalen Koordinaten des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> voraus, gilt nur: Die Kegel sind elliptisch, das Dreieck ist allgemein und die Kreise sind Ellipsen. Inzidenzen und Berührbeziehungen bleiben erhalten. 5) Wählt man, wie bei nicht baryzentrischen homogenen Koordinaten üblich, die Ursprungsebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af7213064a4a766ce7ed1e85449eb0f04f3344d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{3}=0}"> als Ferngerade und setzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {m_{1}}{m_{3}}},y={\tfrac {m_{2}}{m_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {m_{1}}{m_{3}}},y={\tfrac {m_{2}}{m_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd775ad0c3bfd2e2b22f5b5db1679c1b51d44e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.937ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {m_{1}}{m_{3}}},y={\tfrac {m_{2}}{m_{3}}}}">, so beschreibt die Gleichung (SE) im affinen Bereich (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5649fb278f7f259453afc7e9caecf98f6c19ff6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.356ex; height:2.676ex;" alt="{\displaystyle m_{3}\neq 0}">) die Hyperbel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\tfrac {1}{x+1}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\tfrac {1}{x+1}}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be8171322b48a46c2091a2845124fee30189978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.134ex; height:3.676ex;" alt="{\displaystyle y={\tfrac {1}{x+1}}-1}">. In diesem Fall sind die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:0:0),(0:1:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:0:0),(0:1:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83329f8b060f3cb3d5ec77cabc0c39f8c5438c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.376ex; height:2.843ex;" alt="{\displaystyle (1:0:0),(0:1:0)}"> Fernpunkte und zwar die Fernpunkte der Asymptoten. Im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> kann man sich die Hyperbel als Schnittkurve des Kegels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Q}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Q}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b788c6cdc4ae9541835467f094663527492acbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.953ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Q}}_{1}}"> mit der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{3}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17db4fb5d43c4e6f5601b269b97f4afb07c5730d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{3}=1}"> vorstellen. 6) Siehe hierzu auch: <a href="/wiki/Inellipse#Inellipse_und_baryzentrische_Koordinaten" title="Inellipse">Inellipse</a>. <div class="mw-heading mw-heading2"><h2 id="Im_Raum_(n=4,_Tetraeder)">Im Raum (n=4, Tetraeder)</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=20" title="Abschnitt bearbeiten: Im Raum (n=4, Tetraeder)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Im Raum (n=4, Tetraeder)">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Berechnung_und_Eigenschaften">Berechnung und Eigenschaften</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=21" title="Abschnitt bearbeiten: Berechnung und Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Berechnung und Eigenschaften">Quelltext bearbeiten</a>]</div> Im 3-dimensionalen Raum ist ein Simplex ein <a href="/wiki/Tetraeder" title="Tetraeder">Tetraeder</a> mit den Ecken <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},\mathbf {x} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},\mathbf {x} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b593fe5bca264c548c78bea73b49d2f309da40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.963ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},\mathbf {x} _{4}}">. Um die baryzentrischen Koordinaten eines Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> bezgl. des gegebenen Tetraeders zu bestimmen, muss man, analog dem 2-dimensionalen Fall (Dreieck), das homogene lineare Gleichungssystem (siehe Abschnitt Definition) <dl><dd>(G'4)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad a_{1}(\mathbf {x} _{1}-\mathbf {p} )+a_{2}(\mathbf {x} _{2}-\mathbf {p} )+a_{3}(\mathbf {x} _{3}-\mathbf {p} )+a_{4}(\mathbf {x} _{4}-\mathbf {p} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad a_{1}(\mathbf {x} _{1}-\mathbf {p} )+a_{2}(\mathbf {x} _{2}-\mathbf {p} )+a_{3}(\mathbf {x} _{3}-\mathbf {p} )+a_{4}(\mathbf {x} _{4}-\mathbf {p} )=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49df94624557e5d1ead5d92eaf4704d334ae7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.642ex; height:2.843ex;" alt="{\displaystyle \quad a_{1}(\mathbf {x} _{1}-\mathbf {p} )+a_{2}(\mathbf {x} _{2}-\mathbf {p} )+a_{3}(\mathbf {x} _{3}-\mathbf {p} )+a_{4}(\mathbf {x} _{4}-\mathbf {p} )=0}"></dd></dl> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},a_{3},a_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},a_{3},a_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b280beaffed93cad6ce0a0c14d08bd2d1fb11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.238ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},a_{3},a_{4}}"> lösen. Wie im ebenen Fall fügt man hier auch die Normierungsgleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1939c63cb52b782b25b1349e0d6e940383d415da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.918ex; height:2.509ex;" alt="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=1}"> hinzu und löst das LGS mit Hilfe der Cramerschen Regel. Mit den Abkürzungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}={\frac {1}{6}}\det(\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ),\quad V_{2}={\frac {1}{6}}\det(\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}={\frac {1}{6}}\det(\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ),\quad V_{2}={\frac {1}{6}}\det(\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a6a6acd9629d50c5cd41e2299b4783713f9ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.75ex; height:5.176ex;" alt="{\displaystyle V_{1}={\frac {1}{6}}\det(\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ),\quad V_{2}={\frac {1}{6}}\det(\mathbf {x} _{3}-\mathbf {p} ,\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{3}={\frac {1}{6}}\det(\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ),\quad V_{4}={\frac {1}{6}}\det(\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{3}={\frac {1}{6}}\det(\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ),\quad V_{4}={\frac {1}{6}}\det(\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7035c94ee447a8ce27ded69050862782ca0e92a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.103ex; height:5.176ex;" alt="{\displaystyle V_{3}={\frac {1}{6}}\det(\mathbf {x} _{4}-\mathbf {p} ,\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ),\quad V_{4}={\frac {1}{6}}\det(\mathbf {x} _{1}-\mathbf {p} ,\mathbf {x} _{2}-\mathbf {p} ,\mathbf {x} _{3}-\mathbf {p} )}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-tetraeder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Baryzentr-ko-tetraeder.svg/260px-Baryzentr-ko-tetraeder.svg.png" decoding="async" width="260" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Baryzentr-ko-tetraeder.svg/390px-Baryzentr-ko-tetraeder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Baryzentr-ko-tetraeder.svg/520px-Baryzentr-ko-tetraeder.svg.png 2x" data-file-width="249" data-file-height="195" /></a><figcaption>Baryzentrische Koordinaten bezgl. eines Tetraeders (im Raum)</figcaption></figure> erhält man für die baryzentrischen Koordinaten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }">: <dl><dd>(BV4) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (a_{1}:a_{2}:a_{3}:a_{4})=(V_{1}:V_{2}:V_{3}:V_{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (a_{1}:a_{2}:a_{3}:a_{4})=(V_{1}:V_{2}:V_{3}:V_{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381a995498dd06275025ca86f2798e1825c2e3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.695ex; height:2.843ex;" alt="{\displaystyle \ (a_{1}:a_{2}:a_{3}:a_{4})=(V_{1}:V_{2}:V_{3}:V_{4})}"></dd></dl> Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}"> das Volumen des Teiltetraeders, der aus dem gegebenen Tetraeder entsteht, indem man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{i}}"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> ersetzt (s. Bild). Aussage (BV4) ist der Lehrsatz in §25, S. 28, des Buches von Möbius. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd0ee88c6ec7be6311b5650521fa2df97196d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.736ex; height:2.509ex;" alt="{\displaystyle \Delta _{i}}"> die Grundfläche (Seitenfläche des Tetraeders) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"> die Höhe des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ten Teiltetraeders, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}={\tfrac {1}{3}}\Delta _{i}d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}={\tfrac {1}{3}}\Delta _{i}d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f988a62dc0e4c3f0e52ec35d36896f80d2bc50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.656ex; height:3.676ex;" alt="{\displaystyle V_{i}={\tfrac {1}{3}}\Delta _{i}d_{i}}"> und <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}d_{1}:\Delta _{2}d_{2}:\Delta _{3}d_{3}:\Delta _{4}d_{4})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}d_{1}:\Delta _{2}d_{2}:\Delta _{3}d_{3}:\Delta _{4}d_{4})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b667e805c3411c562e4c0a34134a4fbc068943e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.717ex; height:2.843ex;" alt="{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}d_{1}:\Delta _{2}d_{2}:\Delta _{3}d_{3}:\Delta _{4}d_{4})\ .}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Besondere_Punkte">Besondere Punkte</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=22" title="Abschnitt bearbeiten: Besondere Punkte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Besondere Punkte">Quelltext bearbeiten</a>]</div> <dl><dt>Geometrischer Schwerpunkt</dt></dl> Der geometrische Schwerpunkt hat die baryzentrischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1:1:1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1:1:1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ab8a34c1efaee7fe4301bf08a37bd5603c4bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.271ex; height:2.843ex;" alt="{\displaystyle (1:1:1:1)}">. Damit ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}={\frac {1}{3}}\Delta _{i}d_{i}={\frac {1}{4}}V={\frac {1}{4}}{\frac {1}{3}}\Delta _{i}h_{i}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}={\frac {1}{3}}\Delta _{i}d_{i}={\frac {1}{4}}V={\frac {1}{4}}{\frac {1}{3}}\Delta _{i}h_{i}\;,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ffe6f6c722ca2be5c73ba551e5f2fb34faa7220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.142ex; height:5.176ex;" alt="{\displaystyle V_{i}={\frac {1}{3}}\Delta _{i}d_{i}={\frac {1}{4}}V={\frac {1}{4}}{\frac {1}{3}}\Delta _{i}h_{i}\;,}"></dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> das Volumen des gegebenen Tetraeders und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d535f210cbd9b9fe6689e61427b3e213e5b2d547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.139ex; height:2.509ex;" alt="{\displaystyle h_{i}}"> die Höhe des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ten Punktes über dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ten Seitendreieck (s. Bild) ist. Also gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}={\frac {h_{i}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}={\frac {h_{i}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd580792e6f526af9c4797c76ca710b5e8e6641e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.082ex; height:5.343ex;" alt="{\displaystyle d_{i}={\frac {h_{i}}{4}}}"></dd></dl> (Vergleiche die entsprechende Aussage im ebenen Fall.) <dl><dt>Inkugelmittelpunkt</dt></dl> Für den Mittelpunkt der <a href="/wiki/Inkugel" title="Inkugel">Inkugel</a> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}=r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58058fec1b509dd55be4f610eb3b5cdf27d88374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.156ex; height:2.509ex;" alt="{\displaystyle d_{i}=r}"> (Radius der Inkugel) und damit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}:\Delta _{2}:\Delta _{3}:\Delta _{4})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}:\Delta _{2}:\Delta _{3}:\Delta _{4})\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1999caea7074a4abce47b622f055f2a639c176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.017ex; height:2.843ex;" alt="{\displaystyle (a_{1}:a_{2}:a_{3}:a_{4})=(\Delta _{1}:\Delta _{2}:\Delta _{3}:\Delta _{4})\ }"> und</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {3V}{\Delta _{1}+\Delta _{2}+\Delta _{3}+\Delta _{4}}}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>V</mi> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {3V}{\Delta _{1}+\Delta _{2}+\Delta _{3}+\Delta _{4}}}\;,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1bfe07125eed173c96d16ac18918775d03a193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.757ex; height:5.676ex;" alt="{\displaystyle r={\frac {3V}{\Delta _{1}+\Delta _{2}+\Delta _{3}+\Delta _{4}}}\;,}"></dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V_{1}+V_{2}+V_{3}+V_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V_{1}+V_{2}+V_{3}+V_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d976f94a6e176d0c7398ebc8feab1823bf34592d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.045ex; height:2.509ex;" alt="{\displaystyle V=V_{1}+V_{2}+V_{3}+V_{4}}"> das Volumen des gegebenen Tetraeders ist. <dl><dt>Projektion eines Punktes auf eine Koordinatenebene</dt></dl> Analog zum ebenen Fall (siehe oben) ist die Projektion eines Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(\alpha _{1}:\alpha _{2}:\alpha _{3}:\alpha _{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(\alpha _{1}:\alpha _{2}:\alpha _{3}:\alpha _{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b9d1aeb7562493f13d12049c85519566edfd4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.632ex; height:2.843ex;" alt="{\displaystyle P=(\alpha _{1}:\alpha _{2}:\alpha _{3}:\alpha _{4})}"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1:0:0:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1:0:0:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d76067822c3d8ac37c8e6e1ceae8d6d0926466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.348ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1:0:0:0)}"> aus auf die gegenüber liegende Ebene durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2},X_{3},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2},X_{3},X_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b85f83b58ccd565d8813f287a714f2b564b3651d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{2},X_{3},X_{4}}"> (sie hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dac97b493b8da48c509596f28389f8dc2b13853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{1}=0}">) der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;Y_{1}=(0:\alpha _{2}:\alpha _{3}:\alpha _{4})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;Y_{1}=(0:\alpha _{2}:\alpha _{3}:\alpha _{4})\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b3f0c43690a78366f771b56712d54a65ebd756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.202ex; height:2.843ex;" alt="{\displaystyle \;Y_{1}=(0:\alpha _{2}:\alpha _{3}:\alpha _{4})\;}">. Falls die Koordinaten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> normiert sind, teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}Y_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}Y_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee558a5c37b8f39e880a375ffeb64c61e95d34e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.383ex; height:2.509ex;" alt="{\displaystyle X_{1}Y_{1}}"> im Verhältnis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\alpha _{1}):\alpha _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\alpha _{1}):\alpha _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b51afa848ac7a6d3ca4dab452d276b75e7bde6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.833ex; height:2.843ex;" alt="{\displaystyle (1-\alpha _{1}):\alpha _{1}}">. Entsprechendes gilt für die anderen 3 Projektionen. <div class="mw-heading mw-heading3"><h3 id="Satz_von_Commandino">Satz von Commandino</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=23" title="Abschnitt bearbeiten: Satz von Commandino" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=23" title="Quellcode des Abschnitts bearbeiten: Satz von Commandino">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryzentr-ko-tetraeder-ss.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Baryzentr-ko-tetraeder-ss.svg/260px-Baryzentr-ko-tetraeder-ss.svg.png" decoding="async" width="260" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Baryzentr-ko-tetraeder-ss.svg/390px-Baryzentr-ko-tetraeder-ss.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Baryzentr-ko-tetraeder-ss.svg/520px-Baryzentr-ko-tetraeder-ss.svg.png 2x" data-file-width="296" data-file-height="244" /></a><figcaption><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">: Schwerpunkt des Tetraeders, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}">: Schwerpunkte der Dreiecke</figcaption></figure> Projiziert man den geometrischen Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(1:1:1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(1:1:1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293dcf284b61d7056f1d242f6ab9bb7415266760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.868ex; height:2.843ex;" alt="{\displaystyle S=(1:1:1:1)}"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1:0:0:0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1:0:0:0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d76067822c3d8ac37c8e6e1ceae8d6d0926466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.348ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1:0:0:0)}"> aus auf die gegenüberliegende Ebene mit der Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dac97b493b8da48c509596f28389f8dc2b13853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{1}=0}">, erhält man den Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=(0:1:1:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=(0:1:1:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd339cd49591e18c35e26a89896619de4e37814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.848ex; height:2.843ex;" alt="{\displaystyle S_{1}=(0:1:1:1)}"> des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}X_{3}X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}X_{3}X_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f0641ada626e192236c7c0ea1b2b281d309a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.936ex; height:2.509ex;" alt="{\displaystyle X_{2}X_{3}X_{4}}">. Entsprechendes gilt für die anderen Projektionen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">. Also gilt (siehe den vorigen Abschnitt): <dl><dd>Die Gerade durch die Ecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> und den geometrischen Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> des Tetraeders schneidet die gegenüberliegende Dreiecksebene im Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"> des Dreiecks. Dabei teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}S_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fceeffd64681862c04bbb7d3b048ef75ba4e0dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.949ex; height:2.509ex;" alt="{\displaystyle X_{i}S_{i}}"> im Verhältnis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3:1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668c7f7f33ff97edaf55a1d3c7d2e88f810a5575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.262ex; height:2.176ex;" alt="{\displaystyle 3:1}">.</dd></dl> Dies ist der <a href="/wiki/Satz_von_Commandino" title="Satz von Commandino">Satz von Commandino</a>. <div class="mw-heading mw-heading3"><h3 id="Hyperboloid_durch_die_Punkte_eines_Tetraeders">Hyperboloid durch die Punkte eines Tetraeders</h3>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=24" title="Abschnitt bearbeiten: Hyperboloid durch die Punkte eines Tetraeders" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=24" title="Quellcode des Abschnitts bearbeiten: Hyperboloid durch die Punkte eines Tetraeders">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-hyperboloid-tetraeder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Baryko-hyperboloid-tetraeder.svg/220px-Baryko-hyperboloid-tetraeder.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Baryko-hyperboloid-tetraeder.svg/330px-Baryko-hyperboloid-tetraeder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Baryko-hyperboloid-tetraeder.svg/440px-Baryko-hyperboloid-tetraeder.svg.png 2x" data-file-width="259" data-file-height="257" /></a><figcaption>Tetraeder auf einem einschaligen Hyperboloid</figcaption></figure> Ein einschaliges Hyperboloid ist eine Quadrik, die 2 Scharen von Geraden enthält. In geeigneten homogenen Koordinaten kann man es durch die Gleichung <dl><dd>(H)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad a_{1}a_{3}-a_{2}a_{4}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad a_{1}a_{3}-a_{2}a_{4}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ae88bd4666fbdd1f62e070281d3b848cf4b8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.56ex; height:2.509ex;" alt="{\displaystyle \quad a_{1}a_{3}-a_{2}a_{4}=0}"></dd></dl> beschreiben<a href="#cite_note-6">[6]</a> (siehe <a href="/wiki/Einschaliges_Hyperboloid#H1_in_homogenen_Koordinaten" class="mw-redirect" title="Einschaliges Hyperboloid">einschaliges Hyperboloid</a>). Das Hyperboloid enthält die Punkte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=(1:0:0:0),\;X_{2}=(0:1:0:0),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=(1:0:0:0),\;X_{2}=(0:1:0:0),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/085da56428a30c9d87c1d7563a49b21183807df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.021ex; height:2.843ex;" alt="{\displaystyle X_{1}=(1:0:0:0),\;X_{2}=(0:1:0:0),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}=(0:0:1:0),\;X_{4}=(0:0:0:1)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}=(0:0:1:0),\;X_{4}=(0:0:0:1)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31ce18450214832bf8c9fb901e1a6dfa515bea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.602ex; height:2.843ex;" alt="{\displaystyle X_{3}=(0:0:1:0),\;X_{4}=(0:0:0:1)\ .}"></dd></dl> Man rechnet leicht nach, dass <dl><dd>(PH) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad P(u,v)={\big (}(1-u)(1-v):u(1-v):uv:(1-u)v{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>u</mi> <mi>v</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad P(u,v)={\big (}(1-u)(1-v):u(1-v):uv:(1-u)v{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4258f6edaab980a1aa31d137d0658ca0e8e0dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.486ex; height:3.176ex;" alt="{\displaystyle \quad P(u,v)={\big (}(1-u)(1-v):u(1-v):uv:(1-u)v{\big )}}"></dd></dl> eine Parameterdarstellung des Hyperboloids ist. Dabei gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=P(0,0),\ X_{2}=P(1,0),\ X_{3}=P(1,1),\ X_{4}=P(0,1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=P(0,0),\ X_{2}=P(1,0),\ X_{3}=P(1,1),\ X_{4}=P(0,1)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6754c314b24c681c734ae7f8f7d676621ac545f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.387ex; height:2.843ex;" alt="{\displaystyle X_{1}=P(0,0),\ X_{2}=P(1,0),\ X_{3}=P(1,1),\ X_{4}=P(0,1)\ }"> und</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(1:1:1:1)=P({\tfrac {1}{2}},{\tfrac {1}{2}})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(1:1:1:1)=P({\tfrac {1}{2}},{\tfrac {1}{2}})\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aea2ea7f43d8067b45b827d2d04ca4f524a5980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.099ex; height:3.509ex;" alt="{\displaystyle S=(1:1:1:1)=P({\tfrac {1}{2}},{\tfrac {1}{2}})\ .}"></dd></dl> Die Parameterlinien (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}">= const oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">= const) sind Geraden. Da die Summe der baryzentrischen Koordinaten stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> ist, werden allerdings die Punkte des Hyperboloids in der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e7cfd6ac1f746f4e9086b5bc8deb607c31551c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.918ex; height:2.509ex;" alt="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}"> nicht erfasst. Dies ist bei Einführung baryzentrischer Koordinaten kein Nachteil. Fasst man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},a_{3},a_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},a_{3},a_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b280beaffed93cad6ce0a0c14d08bd2d1fb11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.238ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},a_{3},a_{4}}"> als baryzentrische Koordinaten auf, entsprechen die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},X_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbe282436962742f9850b44f04ae273cb0bfe86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.016ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},X_{4}}"> den Ecken eines Tetraeders (in einem affinen Raum) auf einem Hyperboloid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}">, das die Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X_{1}X_{2}}},{\overline {X_{2}X_{3}}},{\overline {X_{3}X_{4}}},{\overline {X_{4}X_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X_{1}X_{2}}},{\overline {X_{2}X_{3}}},{\overline {X_{3}X_{4}}},{\overline {X_{4}X_{1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5825d9e11c96901ac4ede3694f45da18c725dfb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.39ex; height:3.343ex;" alt="{\displaystyle {\overline {X_{1}X_{2}}},{\overline {X_{2}X_{3}}},{\overline {X_{3}X_{4}}},{\overline {X_{4}X_{1}}}}"> enthält (siehe Bild). Die beiden Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X_{2}X_{4}}},{\overline {X_{1}X_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X_{2}X_{4}}},{\overline {X_{1}X_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d1d994707629510e38b38ac5f0193974b6f782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.178ex; height:3.343ex;" alt="{\displaystyle {\overline {X_{2}X_{4}}},{\overline {X_{1}X_{3}}}}"> liegen nicht auf dem Hyperboloid ! Rechnet man die normierten baryzentrischen Koordinaten in affine Koordinaten um (siehe (S) im Abschnitt Definition), erhält man die affine Parameterdarstellung des Hyperboloids: <dl><dd>(APH) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mathbf {p} (u,v)=(1-u)(1-v)\mathbf {x} _{1}+u(1-v)\mathbf {x} _{2}+uv\mathbf {x} _{3}+(1-u)v\mathbf {x} _{4}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>u</mi> <mi>v</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>v</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {p} (u,v)=(1-u)(1-v)\mathbf {x} _{1}+u(1-v)\mathbf {x} _{2}+uv\mathbf {x} _{3}+(1-u)v\mathbf {x} _{4}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb89e83e742b4d61d467972d863bed7eb39430fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.894ex; height:2.843ex;" alt="{\displaystyle \quad \mathbf {p} (u,v)=(1-u)(1-v)\mathbf {x} _{1}+u(1-v)\mathbf {x} _{2}+uv\mathbf {x} _{3}+(1-u)v\mathbf {x} _{4}\ .}"></dd></dl> Dies ist die Darstellung des Hyperboloids als <a href="/wiki/Hyperbolisches_Paraboloid" class="mw-redirect" title="Hyperbolisches Paraboloid">bilineare Interpolationsfläche</a> des räumlichen Vierecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},X_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbe282436962742f9850b44f04ae273cb0bfe86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.016ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},X_{4}}">. <dl><dt>Eigenschaften</dt></dl> Das Hyperboloid hat mit der Fernebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e7cfd6ac1f746f4e9086b5bc8deb607c31551c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.918ex; height:2.509ex;" alt="{\displaystyle a_{1}+a_{2}+a_{3}+a_{4}=0}"> die beiden sich im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(1:-1:1:-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(1:-1:1:-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a17667949f8c585485e206f5a073e6a1a78bca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.728ex; height:2.843ex;" alt="{\displaystyle A=(1:-1:1:-1)}"> schneidenden Geraden <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}:a_{1}+a_{2}=0,\;a_{3}+a_{4}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}:a_{1}+a_{2}=0,\;a_{3}+a_{4}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a310afbd9f53d35d609eaaa54b53d83cd30d513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.765ex; height:2.509ex;" alt="{\displaystyle g_{1}:a_{1}+a_{2}=0,\;a_{3}+a_{4}=0,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}:a_{2}+a_{3}=0,\;a_{1}+a_{4}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}:a_{2}+a_{3}=0,\;a_{1}+a_{4}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5f271bd71ae1035bb5d79a0e65f3c3232100dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.764ex; height:2.509ex;" alt="{\displaystyle g_{2}:a_{2}+a_{3}=0,\;a_{1}+a_{4}=0\;}"></dd></dl> gemeinsam und ist deshalb affin ein <ul><li><a href="/wiki/Hyperbolisches_Paraboloid#H1_in_homogenen_Koordinaten" class="mw-redirect" title="Hyperbolisches Paraboloid">hyperbolisches Paraboloid</a>. (Das obige Bild ist also projektiv zu verstehen.)</li> <li>Die Fernebene ist die Tangentialebene im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.</li> <li>Der Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(1:1:1:1):{\tfrac {1}{4}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3}+\mathbf {x} _{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(1:1:1:1):{\tfrac {1}{4}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3}+\mathbf {x} _{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ba0cf92691ec90cc43d2ace495747953d5f9ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.655ex; height:3.509ex;" alt="{\displaystyle S=(1:1:1:1):{\tfrac {1}{4}}(\mathbf {x} _{1}+\mathbf {x} _{2}+\mathbf {x} _{3}+\mathbf {x} _{4})}"> des Tetraeders liegt auf dem Hyperboloid.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Baryko-tetraeder-koorda.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Baryko-tetraeder-koorda.svg/260px-Baryko-tetraeder-koorda.svg.png" decoding="async" width="260" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Baryko-tetraeder-koorda.svg/390px-Baryko-tetraeder-koorda.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Baryko-tetraeder-koorda.svg/520px-Baryko-tetraeder-koorda.svg.png 2x" data-file-width="259" data-file-height="270" /></a><figcaption>Hyperbolisches Paraboloid (affiner Teil eines projektiven einschaligen Hyperboloids) durch die Ecken eines Tetraeders mit Punkten auf den Koordinatenachsen</figcaption></figure> Die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;g_{3}:a_{1}-a_{3}=0,\;a_{2}-a_{4}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;g_{3}:a_{1}-a_{3}=0,\;a_{2}-a_{4}=0\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e1c1726d5e575025fb446a001850676b11b65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.409ex; height:2.509ex;" alt="{\displaystyle \;g_{3}:a_{1}-a_{3}=0,\;a_{2}-a_{4}=0\;}"> geht durch die Mittelpunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{13}=(1:0:1:0),\;M_{24}=(0:1:0:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{13}=(1:0:1:0),\;M_{24}=(0:1:0:1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ada9a4cf512cb214ba1073996fd4565bec276a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.678ex; height:2.843ex;" alt="{\displaystyle M_{13}=(1:0:1:0),\;M_{24}=(0:1:0:1)}"> der Tetraederkanten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}X_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95642903d6b5b35377714082b0b0b95d73cddd7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{1}X_{3}}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}X_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1868eace910f7b09054a55504338abccad91b32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.957ex; height:2.509ex;" alt="{\displaystyle X_{2}X_{4}}"> und durch den Fernpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(1:-1:1:-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(1:-1:1:-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a17667949f8c585485e206f5a073e6a1a78bca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.728ex; height:2.843ex;" alt="{\displaystyle A=(1:-1:1:-1)}">. Dies bedeutet affin: <ul><li>Die Achsen der Parabeln auf dem hyperbolischen Paraboloid sind alle parallel zur Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207852a9b2705498c65b929b79fdda27b257ec87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}"> durch die Mittelpunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{13}:{\tfrac {1}{2}}(\mathbf {x} _{1}+\mathbf {x} _{3}),\;M_{24}:{\tfrac {1}{2}}(\mathbf {x} _{2}+\mathbf {x} _{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{13}:{\tfrac {1}{2}}(\mathbf {x} _{1}+\mathbf {x} _{3}),\;M_{24}:{\tfrac {1}{2}}(\mathbf {x} _{2}+\mathbf {x} _{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477d445e1f6de3089eab3c8e9406b72265f98085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.29ex; height:3.509ex;" alt="{\displaystyle M_{13}:{\tfrac {1}{2}}(\mathbf {x} _{1}+\mathbf {x} _{3}),\;M_{24}:{\tfrac {1}{2}}(\mathbf {x} _{2}+\mathbf {x} _{4})}"> (siehe <a href="/wiki/Hyperbolisches_Paraboloid#Ebene_Schnitte_von_P2" class="mw-redirect" title="Hyperbolisches Paraboloid">hyperbolisches Paraboloid</a>). Der Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> ist der Mittelpunkt der Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{13},M_{24}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{13},M_{24}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ab4b03dca71811dea85d4daac52105f30e4576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.295ex; height:2.509ex;" alt="{\displaystyle M_{13},M_{24}}">.</li></ul> <dl><dt>Beispiel</dt></dl> Das Bild zeigt das Beispiel mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1}=(1,0,0)^{T},\;\mathbf {x} _{2}=(0,1,0)^{T},\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1}=(1,0,0)^{T},\;\mathbf {x} _{2}=(0,1,0)^{T},\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3f6baad37934fb14002831e4994d994fe30d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.606ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{1}=(1,0,0)^{T},\;\mathbf {x} _{2}=(0,1,0)^{T},\;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{3}=(0,0,1)^{T},\;\mathbf {x} _{4}=(0,0,0)^{T},\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{3}=(0,0,1)^{T},\;\mathbf {x} _{4}=(0,0,0)^{T},\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6817e758367ad1176ee082bd7490e96d6277cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.606ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{3}=(0,0,1)^{T},\;\mathbf {x} _{4}=(0,0,0)^{T},\;}"></dd></dl> Die Parameterdarstellung ist dann <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} (u,v)={\big (}(1-u)(1-v),u(1-v),uv{\big )}^{T}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>u</mi> <mi>v</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} (u,v)={\big (}(1-u)(1-v),u(1-v),uv{\big )}^{T}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ad1d051996b14b4b36383d6cdb7796f99711b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.507ex; height:3.676ex;" alt="{\displaystyle \mathbf {p} (u,v)={\big (}(1-u)(1-v),u(1-v),uv{\big )}^{T}\ .}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Verallgemeinerte_baryzentrische_Koordinaten">Verallgemeinerte baryzentrische Koordinaten</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=25" title="Abschnitt bearbeiten: Verallgemeinerte baryzentrische Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=25" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerte baryzentrische Koordinaten">Quelltext bearbeiten</a>]</div> Baryzentrische Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},\dotsc ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\dotsc ,a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2a623d2e6322e07d9afd0b5cf2566ee55722c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle (a_{1},\dotsc ,a_{n})}">, die mit Bezug auf ein <a href="/wiki/Polytop_(Geometrie)" title="Polytop (Geometrie)">Polytop</a> statt mit Bezug auf ein Simplex definiert sind, werden verallgemeinerte baryzentrische Koordinaten genannt. Hierbei wird weiterhin verlangt, dass die Gleichung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e55ddd7c92a68a6576e76ffcc09eabd2044aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.535ex; height:2.843ex;" alt="{\displaystyle (a_{1}+\dotsb +a_{n})\mathbf {p} =a_{1}\,\mathbf {x} _{1}+\dotsb +a_{n}\,\mathbf {x} _{n}}"></dd></dl> erfüllt wird, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688990fd5b8f4c39c97e473225326f40987148ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.273ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}}"> hier die Eckpunkte des gegebenen Polytops sind. Die Definition ist also formal unverändert, allerdings muss ein Simplex mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> Eckpunkten in einem Vektorraum mit einer Dimension von mindestens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> enthalten sein, während Polytope auch in Vektorräume von niedrigerer Dimension eingebettet sein können. Das einfachste Beispiel ist ein Viereck in der Ebene. Als Konsequenz sind sogar die normierten verallgemeinerten baryzentrischen Koordinaten für ein Polytop im Allgemeinen nicht eindeutig bestimmt, obwohl dies für normierte baryzentrische Koordinaten mit Bezug auf ein Simplex der Fall ist. Verallgemeinerte baryzentrische Koordinaten werden insbesondere in der <a href="/wiki/Computergrafik" title="Computergrafik">Computergrafik</a> und bei der <a href="/wiki/Geometrische_Modellierung" title="Geometrische Modellierung">geometrischen Modellierung</a> verwendet. Dort können dreidimensionale Objekte oft durch Polyeder approximiert werden, sodass die verallgemeinerten baryzentrischen Koordinaten eine geometrische Bedeutung haben und die weitere Bearbeitung dieser Objekte erleichtern. <div class="mw-heading mw-heading2"><h2 id="Baryzentrische_Interpolation">Baryzentrische Interpolation</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=26" title="Abschnitt bearbeiten: Baryzentrische Interpolation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=26" title="Quellcode des Abschnitts bearbeiten: Baryzentrische Interpolation">Quelltext bearbeiten</a>]</div> Auf baryzentrischen Koordinaten basiert ein Interpolationsverfahren, das die <a href="/wiki/Interpolation_(Mathematik)#Allgemeine_lineare_Interpolation" title="Interpolation (Mathematik)">lineare Interpolation</a> für <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> mehrerer Variablen verallgemeinert. Im Falle einer Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> von zwei Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> sind für drei Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x_{A},y_{A})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x_{A},y_{A})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f4bd988e8e73c2da828a7b0afd830092787ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.985ex; height:2.843ex;" alt="{\displaystyle A(x_{A},y_{A})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(x_{B},y_{B})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(x_{B},y_{B})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8841f205dc8d0999c29c8a31f2c15afc179e5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.035ex; height:2.843ex;" alt="{\displaystyle B(x_{B},y_{B})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x_{C},y_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x_{C},y_{C})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eefc6e2433defc2eff9679821c2a66a310bec76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.041ex; height:2.843ex;" alt="{\displaystyle C(x_{C},y_{C})}"> die Funktionswerte gegeben. Dabei dürfen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> nicht auf einer Geraden liegen. Sie müssen also ein Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"> aufspannen. Ist nun ein beliebiger Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> gegeben, so definiert man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=a\,f(x_{A},y_{A})+b\,f(x_{B},y_{B})+c\,f(x_{C},y_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=a\,f(x_{A},y_{A})+b\,f(x_{B},y_{B})+c\,f(x_{C},y_{C})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d16a19e8bace812000a886a58c40189c98fadc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.405ex; height:2.843ex;" alt="{\displaystyle f(x,y)=a\,f(x_{A},y_{A})+b\,f(x_{B},y_{B})+c\,f(x_{C},y_{C})}">,</dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}"> die normierten baryzentrischen Koordinaten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> sind. Diese Interpolation funktioniert auch für Punkte außerhalb des Dreiecks. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=27" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=27" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>Oswin Aichholzer, Bert Jüttler: Einführung in die angewandte Geometrie. Springer-Verlag, Basel 2013, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-0346-0651-6">10.1007/978-3-0346-0651-6</a>, <a href="/wiki/Spezial:ISBN-Suche/9783034606516" class="internal mw-magiclink-isbn">ISBN 978-3-0346-0651-6</a>, S. 59.</li> <li>Gerald Farin, Diane Hansford: Lineare Algebra: Ein geometrischer Zugang. Springer-Verlag, 2013, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-55841-2">10.1007/978-3-642-55841-2</a>, <a href="/wiki/Spezial:ISBN-Suche/9783540418542" class="internal mw-magiclink-isbn">ISBN 978-3-540-41854-2</a>, S. 139.</li> <li>John Fauvel, Raymond Flood, Robin Wilson: Möbius und sein Band: Der Aufstieg von Mathematik und Astronomie im Deutschland des 19. Jahrhunderts. Springer-Verlag, 2013, <a href="/wiki/Spezial:ISBN-Suche/9783034862035" class="internal mw-magiclink-isbn">ISBN 978-3-0348-6203-5</a>, S. 106.</li> <li>Peter Knabner, Lutz Angermann: Numerik partieller Differentialgleichungen. Eine anwendungsorientierte Einführung. Springer 2000, <a href="/wiki/Spezial:ISBN-Suche/3642571816" class="internal mw-magiclink-isbn">ISBN 3-642-57181-6</a>, S. 108–111 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=WdgjBgAAQBAJ&pg=PA108">books.google.de</a>).</li> <li>Abraham A. Ungar: Barycentric Calculus in Euclidean and Hyperbolic Geometry. World Scientific 2010, <a href="/wiki/Spezial:ISBN-Suche/9789814304931" class="internal mw-magiclink-isbn">ISBN 978-981-4304-93-1</a>.</li> <li>John Vince: Mathematics for Computer Graphics. Springer 2010, <a href="/wiki/Spezial:ISBN-Suche/9781849960328" class="internal mw-magiclink-isbn">ISBN 978-1-84996-032-8</a>, S. 208–236.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=28" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=28" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://play.google.com/store/books/details?id=eFPluv_UqFEC">August Ferdinand Möbius: Der baryzentrische Calcul</a></li> <li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BarycentricCoordinates.html">Barycentric Coordinates.</a> In: <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (englisch).</li> <li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Barycentric_coordinates">Barycentric Coordinates.</a> Eintrag in der <a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a>.</li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/barycentriccoordinates">Barycentric Coordinates.</a> Eintrag auf <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li>Philipp B. Laval: <a rel="nofollow" class="external text" href="http://wayback.archive-it.org/all/20131021062259/http://facultyfp.salisbury.edu/despickler/personal/Resources/Graphics/Resources/barycentric.pdf">Mathematics for Computer Graphics – Barycentric Coordinates.</a> (PDF; 137 kB).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/IeM2BYBX">Baryzentrische Koordinaten mit GeoGebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Baryzentrische_Koordinaten&veaction=edit&section=29" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Baryzentrische_Koordinaten&action=edit&section=29" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Max Koecher, Aloys Krieg: Ebene Geometrie. Springer-Verlag, Berlin 2007, <a href="/wiki/Spezial:ISBN-Suche/9783540493280" class="internal mw-magiclink-isbn">ISBN 978-3-540-49328-0</a>, S. 76. </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> August Ferdinand Möbius: Der baryzentrische Calcul, Verlag von Johann Ambrosius BartH, Leipzig, 1827. </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> Josef Hoschek, Dieter Lasser: Grundlagen der geometriechen Datenverarbeitung. Teubner-Verlag,, 1989, <a href="/wiki/Spezial:ISBN-Suche/3519029626" class="internal mw-magiclink-isbn">ISBN 3-519-02962-6</a>, S. 243. </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> Gerald Farin: Curves and Surfeces for Computer Aided Geometric Design. Academic Press, 1990, <a href="/wiki/Spezial:ISBN-Suche/0122490517" class="internal mw-magiclink-isbn">ISBN 0-12-249051-7</a>, S. 20. </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Christian Gerthsen: Physik. Springer-Verlag, 1963, S. 37. </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> Felix Klein: Vorlesungen über höhere Geometrie, Springer-Verlag, 2013 <a href="/wiki/Spezial:ISBN-Suche/3642886744" class="internal mw-magiclink-isbn">ISBN 3642886744</a>, 9783642886744, S. 15. </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Baryzentrische_Koordinaten&oldid=245101595">https://de.wikipedia.org/w/index.php?title=Baryzentrische_Koordinaten&oldid=245101595</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Lineare_Algebra" title="Kategorie:Lineare Algebra">Lineare Algebra</a></li><li><a href="/wiki/Kategorie:Koordinatensystem" title="Kategorie:Koordinatensystem">Koordinatensystem</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Baryzentrische+Koordinaten" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Barysentrinen_koordinaatti_(geometria)" title="Barysentrinen koordinaatti (geometria) – Finnisch" lang="fi" hreflang="fi" data-title="Barysentrinen koordinaatti (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Coordonn%C3%A9es_barycentriques" title="Coordonnées barycentriques – Französisch" lang="fr" hreflang="fr" data-title="Coordonnées barycentriques" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coordenadas_baric%C3%A9ntricas" title="Coordenadas baricéntricas – Galicisch" lang="gl" hreflang="gl" data-title="Coordenadas baricéntricas" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%91%D7%A8%D7%99%D7%A6%D7%A0%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="קואורדינטות בריצנטריות – Hebräisch" lang="he" hreflang="he" data-title="קואורדינטות בריצנטריות" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coordinate_baricentriche" title="Coordinate baricentriche – Italienisch" lang="it" hreflang="it" data-title="Coordinate baricentriche" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Barycentrische_co%C3%B6rdinaten" title="Barycentrische coördinaten – Niederländisch" lang="nl" hreflang="nl" data-title="Barycentrische coördinaten" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Barysentriske_koordinater" title="Barysentriske koordinater – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Barysentriske koordinater" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wsp%C3%B3%C5%82rz%C4%99dne_barycentryczne_(matematyka)" title="Współrzędne barycentryczne (matematyka) – Polnisch" lang="pl" hreflang="pl" data-title="Współrzędne barycentryczne (matematyka)" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coordenadas_baric%C3%AAntricas" title="Coordenadas baricêntricas – Portugiesisch" lang="pt" hreflang="pt" data-title="Coordenadas baricêntricas" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%8B" title="Барицентрические координаты – Russisch" lang="ru" hreflang="ru" data-title="Барицентрические координаты" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Te%C5%BEi%C5%A1%C4%8Dni_koordinatni_sistem" title="Težiščni koordinatni sistem – Slowenisch" lang="sl" hreflang="sl" data-title="Težiščni koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Barycentriska_koordinater" title="Barycentriska koordinater – Schwedisch" lang="sv" hreflang="sv" data-title="Barycentriska koordinater" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%BD%D1%96_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B8" title="Барицентричні координати – Ukrainisch" lang="uk" hreflang="uk" data-title="Барицентричні координати" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%8Da_%C4%91%E1%BB%99_t%E1%BB%89_c%E1%BB%B1" title="Tọa độ tỉ cự – Vietnamesisch" lang="vi" hreflang="vi" data-title="Tọa độ tỉ cự" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%87%8D%E5%BF%83%E5%9D%90%E6%A0%87" title="重心坐标 – Chinesisch" lang="zh" hreflang="zh" data-title="重心坐标" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8D%E5%BF%83%E5%BA%A7%E6%A8%99" title="重心座標 – Kantonesisch" lang="yue" hreflang="yue" data-title="重心座標" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q738422#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 19. 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Baryzentrische Koordinaten – Wikipedia