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Chr. – 190 v. Chr.), beschreibt eine Flächengleichheit bei einem beliebigen <a href="/wiki/Dreieck" title="Dreieck">Dreieck</a> mit zugehöriger <a href="/wiki/Seitenhalbierende" title="Seitenhalbierende">Seitenhalbierenden</a>. </p><p>Sei <i>ABC</i> ein Dreieck und <i>AD</i> die Seitenhalbierende der Seite <i>BC</i>, dann gilt die folgende Gleichung: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |AB|^{2}+|AC|^{2}=2(|AD|^{2}+|BD|^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>D</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>D</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |AB|^{2}+|AC|^{2}=2(|AD|^{2}+|BD|^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c0e27127bda0b5f03265314d2799144cafd0f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.515ex; height:3.343ex;" alt="{\displaystyle |AB|^{2}+|AC|^{2}=2(|AD|^{2}+|BD|^{2})}" /></span></dd></dl> <p>Der Satz ergibt sich auch als ein Spezialfall des <a href="/wiki/Satz_von_Stewart" title="Satz von Stewart">Satzes von Stewart</a>, der eine Gleichung für den Fall einer beliebigen <a href="/wiki/Cevane" title="Cevane">Cevane</a> anstatt einer Seitenhalbierenden liefert. Im Gegensatz zum Satz von Apollonios lassen sich die im Satz von Stewart auftretenden Größen jedoch nicht direkt als Flächen deuten. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Appolonius_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Appolonius_theorem.svg/260px-Appolonius_theorem.svg.png" decoding="async" width="260" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Appolonius_theorem.svg/390px-Appolonius_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Appolonius_theorem.svg/520px-Appolonius_theorem.svg.png 2x" data-file-width="549" data-file-height="652" /></a><figcaption>Pythagoras als Spezialfall:<br /> grüne Fläche = rote Fläche</figcaption></figure> <p>Besitzt das Dreieck <i>ABC</i> einen rechten Winkel in <i>A</i>, so erhält man aus dem Satz des Apollonios den <a href="/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a> als <a href="/wiki/Korollar" title="Korollar">Korollar</a>, da die Seitenhalbierende <i>AD</i> dann dem Radius des zugehörigen <a href="/wiki/Thaleskreis" class="mw-redirect" title="Thaleskreis">Thaleskreises</a> entspricht. Das heißt, es gilt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |AD|=|BD|=|CD|={\frac {|BC|}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |AD|=|BD|=|CD|={\frac {|BC|}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f4313798d2cef62ffdd9bf1db6c19f7bfe3ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.883ex; height:5.676ex;" alt="{\displaystyle |AD|=|BD|=|CD|={\frac {|BC|}{2}}}" /></span></dd></dl> <p>Damit ergibt sich dann der Satz des Pythagoras: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |AB|^{2}+|AC|^{2}=2\left(\left({\frac {|BC|}{2}}\right)^{2}+\left({\frac {|BC|}{2}}\right)^{2}\right)=|BC|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |AB|^{2}+|AC|^{2}=2\left(\left({\frac {|BC|}{2}}\right)^{2}+\left({\frac {|BC|}{2}}\right)^{2}\right)=|BC|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0023570e8eadb1d4c1e1ed2ae4b216fae8a63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.97ex; height:7.509ex;" alt="{\displaystyle |AB|^{2}+|AC|^{2}=2\left(\left({\frac {|BC|}{2}}\right)^{2}+\left({\frac {|BC|}{2}}\right)^{2}\right)=|BC|^{2}}" /></span></dd></dl> <p>Verdoppelt man das Dreieck <i>ABC</i> zu einem Parallelogramm, so dass die Seite <i>BC</i> eine der Diagonalen des Parallelogramms wird, dann liefert der Satz des Apollonios die <a href="/wiki/Parallelogrammgleichung" title="Parallelogrammgleichung">Parallelogrammgleichung</a>. Außerhalb der Elementargeometrie spricht man dementsprechend auch von der <a href="/wiki/Apollonios-Gleichung" title="Apollonios-Gleichung">Apollonios-Gleichung</a>. </p><p>Es gilt auch die Umkehrung des Satzes von Apollonios, das heißt, ist für einen beliebigen Punkt <i>D</i> auf <i>BC</i> die obige Gleichung erfüllt, so ist <i>D</i> der Mittelpunkt von <i>BC</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Satz_von_Apollonios_(Dreieck)&veaction=edit&section=1" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Satz_von_Apollonios_(Dreieck)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Claudi Alsina, Roger B. Nelsen: <i>Perlen der Mathematik: 20 geometrische Figuren als Ausgangspunkte für mathematische Erkundungsreisen</i>. Springer, 2015, <a href="/wiki/Spezial:ISBN-Suche/9783662454619" class="internal mw-magiclink-isbn">ISBN 978-3-662-45461-9</a>, S. 63 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=4DavMl7-aFgC&pg=PA65">Auszug aus der englischen Ausgabe</a> books.google.de)</li> <li>A. J. G. May: <i>1535. The Converse of Apollonius’ Theorem</i>. In: <i>The Mathematical Gazette</i>, Band 25, Nr. 266 (Okt., 1941), S. 228–229 (<a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/3606591">3606591</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Satz_von_Apollonios_(Dreieck)&veaction=edit&section=2" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Satz_von_Apollonios_(Dreieck)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Peter Gallin: <a rel="nofollow" class="external text" href="https://www.gallin.ch/091102GallinLayoutBulletin.pdf"><i>Ein Theorem von Apollonius</i>.</a> (PDF)</li> <li>David B. 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Französisch" lang="fr" hreflang="fr" data-title="Théorème de la médiane" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%AA%D7%99%D7%9B%D7%95%D7%9F" title="משפט התיכון – Hebräisch" lang="he" hreflang="he" data-title="משפט התיכון" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A5%8B%E0%A4%B2%E0%A5%8B%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%B8_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="अपोलोनियस का प्रमेय – Hindi" lang="hi" hreflang="hi" data-title="अपोलोनियस का प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%BA%D5%B8%D5%AC%D5%B8%D5%B6%D5%AB%D5%B8%D6%82%D5%BD%D5%AB_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Ապոլոնիուսի թեորեմ – Armenisch" lang="hy" hreflang="hy" data-title="Ապոլոնիուսի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_Apollonius" title="Teorema Apollonius – Indonesisch" lang="id" hreflang="id" data-title="Teorema Apollonius" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_della_mediana" title="Teorema della mediana – Italienisch" lang="it" hreflang="it" data-title="Teorema della mediana" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%9D%E3%83%AD%E3%83%8B%E3%82%A6%E3%82%B9%E3%81%AE%E5%AE%9A%E7%90%86" title="アポロニウスの定理 – Japanisch" lang="ja" hreflang="ja" data-title="アポロニウスの定理" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%91%E1%9F%92%E1%9E%9A%E1%9E%B9%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%B8%E1%9E%94%E1%9E%91%E1%9E%98%E1%9F%81%E1%9E%8A%E1%9F%92%E1%9E%99%E1%9E%B6%E1%9E%93" title="ទ្រឹស្តីបទមេដ្យាន – Khmer" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទមេដ្យាន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%ED%8F%B4%EB%A1%9C%EB%8B%88%EC%98%A4%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="아폴로니오스 정리 – Koreanisch" lang="ko" hreflang="ko" data-title="아폴로니오스 정리" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BF%D0%BE%D0%BB%D0%BB%D0%BE%D0%BD%D1%8B_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC" title="Аполлоны теорем – Mongolisch" lang="mn" hreflang="mn" data-title="Аполлоны теорем" data-language-autonym="Монгол" data-language-local-name="Mongolisch" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Apollonius" title="Stelling van Apollonius – Niederländisch" lang="nl" hreflang="nl" data-title="Stelling van Apollonius" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/To%C5%BCsamo%C5%9B%C4%87_Apoloniusza" title="Tożsamość Apoloniusza – Polnisch" lang="pl" hreflang="pl" data-title="Tożsamość Apoloniusza" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_dla_median-a" title="Teorema dla median-a – Piemontesisch" lang="pms" hreflang="pms" data-title="Teorema dla median-a" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_medianei" title="Teorema medianei – Rumänisch" lang="ro" hreflang="ro" data-title="Teorema medianei" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%90%D0%BF%D0%BE%D0%BB%D0%BB%D0%BE%D0%BD%D0%B8%D1%8F" title="Теорема Аполлония – Russisch" lang="ru" hreflang="ru" data-title="Теорема Аполлония" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Apolonijev_izrek" title="Apolonijev izrek – Slowenisch" lang="sl" hreflang="sl" data-title="Apolonijev izrek" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Apollonios_sats" title="Apollonios sats – Schwedisch" lang="sv" hreflang="sv" data-title="Apollonios sats" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%8A%E0%AE%B2%E0%AF%8B%E0%AE%A9%E0%AE%BF%E0%AE%AF%E0%AE%9A%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="அப்பொலோனியசின் தேற்றம் – Tamil" lang="ta" hreflang="ta" data-title="அப்பொலோனியசின் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B8%B0%E0%B8%9E%E0%B8%AD%E0%B8%A5%E0%B9%82%E0%B8%A5%E0%B9%80%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B8%AA" title="ทฤษฎีบทของอะพอลโลเนียส – Thailändisch" lang="th" hreflang="th" data-title="ทฤษฎีบทของอะพอลโลเนียส" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Apollonius_teoremi" title="Apollonius teoremi – Türkisch" lang="tr" hreflang="tr" data-title="Apollonius teoremi" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%90%D0%BF%D0%BE%D0%BB%D0%BB%D0%BE%D0%BD%D1%96%D1%8F" title="Теорема Аполлонія – Ukrainisch" lang="uk" hreflang="uk" data-title="Теорема Аполлонія" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Apollonius" title="Định lý Apollonius – Vietnamesisch" lang="vi" hreflang="vi" data-title="Định lý Apollonius" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E7%B7%9A%E5%AE%9A%E7%90%86" title="中線定理 – Chinesisch" lang="zh" hreflang="zh" data-title="中線定理" data-language-autonym="中文" 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