Ορθόκεντρο τριγώνου - Βικιπαίδεια

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mw-portlet-personal user-links-collapsible-item" title="Μενού χρήστη" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_el.wikipedia.org&uselang=el"><span>Δωρεές</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%94%CE%B7%CE%BC%CE%B9%CE%BF%CF%85%CF%81%CE%B3%CE%AF%CE%B1%CE%9B%CE%BF%CE%B3%CE%B1%CF%81%CE%B9%CE%B1%CF%83%CE%BC%CE%BF%CF%8D&returnto=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Σας ενθαρρύνουμε να δημιουργήσετε ένα λογαριασμό και να συνδεθείτε· ωστόσο, δεν είναι υποχρεωτικό"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Δημιουργία 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vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Περιεχόμενα</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">μετακίνηση στην πλαϊνή μπάρα</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">απόκρυψη</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Αρχή</div> </a> </li> <li id="toc-Αποδείξεις" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Αποδείξεις"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Αποδείξεις</span> </div> </a> <ul id="toc-Αποδείξεις-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ιδιότητες" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ιδιότητες"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ιδιότητες</span> </div> </a> <ul id="toc-Ιδιότητες-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Μετρικές_σχέσεις" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Μετρικές_σχέσεις"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Μετρικές σχέσεις</span> </div> </a> <ul id="toc-Μετρικές_σχέσεις-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Δείτε_επίσης" class="vector-toc-list-item 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class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Περιεχόμενα" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Εναλλαγή του πίνακα περιεχομένων" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Εναλλαγή του πίνακα περιεχομένων</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ορθόκεντρο τριγώνου</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Μεταβείτε σε ένα λήμμα σε άλλη γλώσσα. Διαθέσιμο σε 21 γλώσσες" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 γλώσσες</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D1%8A%D1%80" title="Ортоцентър – Βουλγαρικά" lang="bg" hreflang="bg" data-title="Ортоцентър" data-language-autonym="Български" data-language-local-name="Βουλγαρικά" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ortocentre" title="Ortocentre – Καταλανικά" lang="ca" hreflang="ca" data-title="Ortocentre" data-language-autonym="Català" data-language-local-name="Καταλανικά" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D1%80" title="Ортоцентр – Τσουβασικά" lang="cv" hreflang="cv" data-title="Ортоцентр" data-language-autonym="Чӑвашла" data-language-local-name="Τσουβασικά" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/H%C3%B6henschnittpunkt" title="Höhenschnittpunkt – Γερμανικά" lang="de" hreflang="de" data-title="Höhenschnittpunkt" data-language-autonym="Deutsch" data-language-local-name="Γερμανικά" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Orthocenter" title="Orthocenter – Αγγλικά" lang="en" hreflang="en" data-title="Orthocenter" data-language-autonym="English" data-language-local-name="Αγγλικά" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ortocentro" title="Ortocentro – Ισπανικά" lang="es" hreflang="es" data-title="Ortocentro" data-language-autonym="Español" data-language-local-name="Ισπανικά" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ortozentro" title="Ortozentro – Βασκικά" lang="eu" hreflang="eu" data-title="Ortozentro" data-language-autonym="Euskara" data-language-local-name="Βασκικά" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ortokeskus" title="Ortokeskus – Φινλανδικά" lang="fi" hreflang="fi" data-title="Ortokeskus" data-language-autonym="Suomi" data-language-local-name="Φινλανδικά" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ortocentro" title="Ortocentro – Γαλικιανά" lang="gl" hreflang="gl" data-title="Ortocentro" data-language-autonym="Galego" data-language-local-name="Γαλικιανά" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ortocentar" title="Ortocentar – Κροατικά" lang="hr" hreflang="hr" data-title="Ortocentar" data-language-autonym="Hrvatski" data-language-local-name="Κροατικά" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%95%D6%80%D5%A9%D5%B8%D5%AF%D5%A5%D5%B6%D5%BF%D6%80%D5%B8%D5%B6" title="Օրթոկենտրոն – Αρμενικά" lang="hy" hreflang="hy" data-title="Օրթոկենտրոն" data-language-autonym="Հայերեն" data-language-local-name="Αρμενικά" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ortocentro" title="Ortocentro – Ιταλικά" lang="it" hreflang="it" data-title="Ortocentro" data-language-autonym="Italiano" data-language-local-name="Ιταλικά" 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/%E5%9E%82%E5%BF%83" title="垂心 – Ιαπωνικά" lang="ja" hreflang="ja" data-title="垂心" data-language-autonym="日本語" data-language-local-name="Ιαπωνικά" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D1%80" title="Ортоцентр – Καζακικά" lang="kk" hreflang="kk" data-title="Ортоцентр" data-language-autonym="Қазақша" data-language-local-name="Καζακικά" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EC%8B%AC_(%EA%B8%B0%ED%95%98%ED%95%99)" title="수심 (기하학) – Κορεατικά" lang="ko" hreflang="ko" data-title="수심 (기하학)" data-language-autonym="한국어" data-language-local-name="Κορεατικά" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoogtepunt_(meetkunde)" title="Hoogtepunt (meetkunde) – Ολλανδικά" lang="nl" hreflang="nl" data-title="Hoogtepunt (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Ολλανδικά" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ortocentro" title="Ortocentro – Πορτογαλικά" lang="pt" hreflang="pt" data-title="Ortocentro" data-language-autonym="Português" data-language-local-name="Πορτογαλικά" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D1%80" title="Ортоцентр – Ρωσικά" lang="ru" hreflang="ru" data-title="Ортоцентр" data-language-autonym="Русский" data-language-local-name="Ρωσικά" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ortocentrum" title="Ortocentrum – Σουηδικά" lang="sv" hreflang="sv" data-title="Ortocentrum" data-language-autonym="Svenska" 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title="Γεωμετρία">γεωμετρία</a>, το <b>ορθόκεντρο</b> ενός <a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο">τριγώνου</a> είναι το σημείο που τέμνονται τα <a href="/wiki/%CE%8E%CF%88%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Ύψος τριγώνου">ύψη</a> του τριγώνου (ή οι προεκτάσεις τους).<sup id="cite_ref-Tav_1-0" class="reference"><a href="#cite_note-Tav-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-T57_2-0" class="reference"><a href="#cite_note-T57-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A75_3-0" class="reference"><a href="#cite_note-A75-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-P74_4-0" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Σε ένα <a href="/wiki/%CE%9F%CE%BE%CF%85%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Οξυγώνιο τρίγωνο">οξυγώνιο τρίγωνο</a>, το ορθόκεντρο είναι εσωτερικό σημείο του τριγώνου, σε ένα <a href="/wiki/%CE%91%CE%BC%CE%B2%CE%BB%CF%85%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Αμβλυγώνιο τρίγωνο">αμβλυγώνιο τρίγωνο</a> είναι εξωτερικό και σε ένα <a href="/wiki/%CE%9F%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ορθογώνιο τρίγωνο">ορθογώνιο τρίγωνο</a> ταυτίζεται με την κορυφή που αντιστοιχεί στην <a href="/wiki/%CE%9F%CF%81%CE%B8%CE%AE_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1" title="Ορθή γωνία">ορθή γωνία</a>. </p> <div class="thumb tmulti tnone center"><div class="thumbinner" style="width:612px;max-width:612px"><div class="tsingle" style="float:left;margin:1px;width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Acute_triangle_orthocenter_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Acute_triangle_orthocenter_el.svg/200px-Acute_triangle_orthocenter_el.svg.png" decoding="async" width="200" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Acute_triangle_orthocenter_el.svg/300px-Acute_triangle_orthocenter_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Acute_triangle_orthocenter_el.svg/400px-Acute_triangle_orthocenter_el.svg.png 2x" data-file-width="178" data-file-height="122" /></a></span></div></div><div class="tsingle" style="float:left;margin:1px;width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Right_triangle_orthocenter_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Right_triangle_orthocenter_el.svg/200px-Right_triangle_orthocenter_el.svg.png" decoding="async" width="200" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Right_triangle_orthocenter_el.svg/300px-Right_triangle_orthocenter_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Right_triangle_orthocenter_el.svg/400px-Right_triangle_orthocenter_el.svg.png 2x" data-file-width="151" data-file-height="87" /></a></span></div></div><div class="tsingle" style="float:left;margin:1px;width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Obtuse_triangle_orthocenter_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Obtuse_triangle_orthocenter_el.svg/200px-Obtuse_triangle_orthocenter_el.svg.png" decoding="async" width="200" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Obtuse_triangle_orthocenter_el.svg/300px-Obtuse_triangle_orthocenter_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Obtuse_triangle_orthocenter_el.svg/400px-Obtuse_triangle_orthocenter_el.svg.png 2x" data-file-width="151" data-file-height="122" /></a></span></div></div><div style="clear:left"></div><div class="thumbcaption" style="clear:left;text-align:left;background-color:transparent">Το ορθόκεντρο <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 {\rm {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d5d424a20363b5b429c22bf0fa9b5eb612c429" 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 {\rm {H}}}"></span> σε ένα <a href="/wiki/%CE%9F%CE%BE%CF%85%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Οξυγώνιο τρίγωνο">οξυγώνιο</a>, ένα <a href="/wiki/%CE%9F%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ορθογώνιο τρίγωνο">ορθογώνιο</a> και ένα <a href="/wiki/%CE%91%CE%BC%CE%B2%CE%BB%CF%85%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Αμβλυγώνιο τρίγωνο">αμβλυγώνιο</a> τρίγωνο.</div></div></div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Αποδείξεις"><span id=".CE.91.CF.80.CE.BF.CE.B4.CE.B5.CE.AF.CE.BE.CE.B5.CE.B9.CF.82"></span>Αποδείξεις</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=1" title="Επεξεργασία ενότητας: Αποδείξεις" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=1" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Αποδείξεις"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r9961950">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Θεώρημα</strong><span class="theoreme-tiret"> — </span> Σε κάθε τρίγωνο <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 \mathrm {AB\Gamma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AB\Gamma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612e145b375b84f995485f37d1bbfe01e1f81700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle \mathrm {AB\Gamma } }"></span>, τα ύψη <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 \upsilon _{\alpha },\upsilon _{\beta },\upsilon _{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \upsilon _{\alpha },\upsilon _{\beta },\upsilon _{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07dd0c9446b922dff3135f0548c36941277ae07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.417ex; height:2.343ex;" alt="{\displaystyle \upsilon _{\alpha },\upsilon _{\beta },\upsilon _{\gamma }}"></span> (ή οι προεκτάσεις τους) διέρχονται από το ίδιο σημείο. </p> </div> <style data-mw-deduplicate="TemplateStyles:r10213552">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style> <table role="presentation" class="math_proof" style="display: block"> <tbody><tr> <td><strong>Απόδειξη (με αντισυμπληρωματικό τρίγωνο)</strong>   <b>[</b> <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/4/41/Orthocenter_step_by_step_proof_using_antimedial_el.svg">Βήμα προς βήμα</a> <b>]</b>   </td></tr> <tr> <td> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Orthocenter_proof_using_anti_median_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Orthocenter_proof_using_anti_median_triangle.svg/220px-Orthocenter_proof_using_anti_median_triangle.svg.png" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Orthocenter_proof_using_anti_median_triangle.svg/330px-Orthocenter_proof_using_anti_median_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Orthocenter_proof_using_anti_median_triangle.svg/440px-Orthocenter_proof_using_anti_median_triangle.svg.png 2x" data-file-width="185" data-file-height="141" /></a><figcaption>Τα ύψη του <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span> είναι οι μεσοκάθετοι του <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 {\rm {K\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2f91d124b5f9e88a70b12dfddb162fd5f73024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda M}}}"></span>.</figcaption></figure> <p>Έστω τρίγωνο <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span>. Θα κατασκευάσουμε ένα τρίγωνο <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 {\rm {K\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2f91d124b5f9e88a70b12dfddb162fd5f73024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda M}}}"></span> ώστε οι <a href="/wiki/%CE%9C%CE%B5%CF%83%CE%BF%CE%BA%CE%AC%CE%B8%CE%B5%CF%84%CE%B7_%CE%B5%CF%85%CE%B8%CF%8D%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF%CF%85_%CF%84%CE%BC%CE%AE%CE%BC%CE%B1%CF%84%CE%BF%CF%82" title="Μεσοκάθετη ευθύγραμμου τμήματος">μεσοκάθετοι</a> των πλευρών του να συμπίπτουν με τα ύψη του <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span>. Για τις μεσοκαθέτους γνωρίζουμε ότι διέρχονται από το ίδιο σημείο, το <a href="/wiki/%CE%A0%CE%B5%CF%81%CE%AF%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF" class="mw-redirect" title="Περίκεντρο">περίκεντρο</a>, και έτσι θα καταλήξουμε ότι και τα ύψη του <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span> (ή οι προεκτάσεις τους) διέρχονται από το ίδιο σημείο. </p><p>Θεωρούμε την ευθεία που διέρχεται από το <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 {\rm {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d483866242c3b3266289fb4d3bdd0b3b947863e" 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 {\rm {A}}}"></span> και είναι παράλληλη στο <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 {\rm {B\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376c40198ce1f0b8abb3582720709e7bfb986d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\rm {B\Gamma }}}"></span>, την ευθεία που διέρχεται από το <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 {\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881845848d4307e1e31581818b3ea87ff05c5cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\rm {B}}}"></span> και είναι παράλληλη στο <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 {\rm {A\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319ed750ea4cf213d690a02f8f7af28709d80d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.196ex; height:2.176ex;" alt="{\displaystyle {\rm {A\Gamma }}}"></span> και την ευθεία που διέρχεται από το <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 {\rm {\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33203cfe0d0b5b98ffc39d1d97ed2817407ac47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle {\rm {\Gamma }}}"></span> και είναι παράλληλη στο <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 {\rm {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc35a324d3e970023abcb065da5264b1ba1e0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.389ex; height:2.176ex;" alt="{\displaystyle {\rm {AB}}}"></span>. Έστω <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 {\rm {K\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2f91d124b5f9e88a70b12dfddb162fd5f73024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda M}}}"></span> το τρίγωνο που σχηματίζουν αυτές οι τρεις ευθείες.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>Σημείωση 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>Σημείωση 2<span class="cite-bracket">]</span></a></sup> </p><p>Το τετράπλευρο <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 {\rm {KAB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {KAB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189c248dbc8970ab2f0459770b4f280444fe8f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.65ex; height:2.176ex;" alt="{\displaystyle {\rm {KAB\Gamma }}}"></span> είναι <a href="/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CE%BB%CE%BB%CE%B7%CE%BB%CF%8C%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF" title="Παραλληλόγραμμο">παραλληλόγραμμο</a> καθώς οι πλευρές του είναι παράλληλες, επομένως </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 {\rm {AK}}={\rm {B\Gamma }}=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AK}}={\rm {B\Gamma }}=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8a2eae7fe9a03721939297d9bc5bc47066af0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.334ex; height:2.176ex;" alt="{\displaystyle {\rm {AK}}={\rm {B\Gamma }}=\alpha }"></span>.</dd></dl> <p>Αντίστοιχα, από το παραλληλόγραμμο <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 {\rm {A\Lambda \Gamma B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Lambda \Gamma B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ca4f176947c5895b58e27c74f241cc5c730f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.454ex; height:2.176ex;" alt="{\displaystyle {\rm {A\Lambda \Gamma B}}}"></span> έχουμε ότι </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 {\rm {\Lambda A}}={\rm {B\Gamma }}=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Lambda A}}={\rm {B\Gamma }}=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92b65a91c801b2a94641007bb121e4587813971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.139ex; height:2.176ex;" alt="{\displaystyle {\rm {\Lambda A}}={\rm {B\Gamma }}=\alpha }"></span>.</dd></dl> <p>Συνεπώς το <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 {\rm {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d483866242c3b3266289fb4d3bdd0b3b947863e" 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 {\rm {A}}}"></span> είναι το μέσο του <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 {\rm {K\Lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8956a51785ccdbcbafff64ef12bccd02b1a95504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.421ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda }}}"></span> και η μεσοκάθετος του <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 {\rm {K\Lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8956a51785ccdbcbafff64ef12bccd02b1a95504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.421ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda }}}"></span> διέρχεται από το <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 {\rm {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d483866242c3b3266289fb4d3bdd0b3b947863e" 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 {\rm {A}}}"></span>. Επίσης είναι κάθετη στο <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 {\rm {B\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376c40198ce1f0b8abb3582720709e7bfb986d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\rm {B\Gamma }}}"></span> (καθώς <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 {\rm {K\Lambda }}\parallel {\rm {B\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> </mrow> </mrow> <mo>∥</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda }}\parallel {\rm {B\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabfc86dcbf5d45d694be28641828f722dc82b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.972ex; height:2.843ex;" alt="{\displaystyle {\rm {K\Lambda }}\parallel {\rm {B\Gamma }}}"></span>), άρα το ύψος <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 \upsilon _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \upsilon _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50dd1bac3c7ea8d98bd8b88257a23e1ba4bf362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \upsilon _{\alpha }}"></span> ανήκει σε αυτή. </p><p>Αντίστοιχα, τα <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 {\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881845848d4307e1e31581818b3ea87ff05c5cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\rm {B}}}"></span> και <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 {\rm {\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33203cfe0d0b5b98ffc39d1d97ed2817407ac47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle {\rm {\Gamma }}}"></span> είναι τα μέσα των <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 {\rm {\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23be6e392c2891425d9e3859d7c6609d0146f9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle {\rm {\Lambda M}}}"></span> και <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 {\rm {KM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {KM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1744229cbe45c0667ffb173e58ed6baeaece57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.939ex; height:2.176ex;" alt="{\displaystyle {\rm {KM}}}"></span>, και τα <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 \upsilon _{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \upsilon _{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e3e475cdff1be8800bcc768233f4afbe008e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.429ex; height:2.343ex;" alt="{\displaystyle \upsilon _{\beta }}"></span> και <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 \upsilon _{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \upsilon _{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0442faafaf4ea81a23ef88f887daae98716e6d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.38ex; height:2.343ex;" alt="{\displaystyle \upsilon _{\gamma }}"></span> ανήκουν στις μεσοκαθέτους των <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 {\rm {\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23be6e392c2891425d9e3859d7c6609d0146f9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle {\rm {\Lambda M}}}"></span> και <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 {\rm {KM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {KM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1744229cbe45c0667ffb173e58ed6baeaece57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.939ex; height:2.176ex;" alt="{\displaystyle {\rm {KM}}}"></span>. Συνεπώς, καταλήγουμε ότι τα ύψη (ή οι προεκτάσεις τους) διέρχονται από το περίκεντρο του <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 {\rm {K\Lambda M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">Λ</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {K\Lambda M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2f91d124b5f9e88a70b12dfddb162fd5f73024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle {\rm {K\Lambda M}}}"></span>. </p> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10213552"> <table role="presentation" class="math_proof {{ safesubst:p{{ safesubst:#iftrue: mw-collapsible mw-collapsed |1|2}}| |}}" style="display: block"> <tbody><tr> <td><strong>Απόδειξη (με εγγεγραμμένα τετράπλευρα)</strong>   <b>[</b> <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/a/a1/Orthocenter_step_by_step_proof_el.svg">Βήμα προς βήμα</a> <b>]</b>   </td></tr> <tr> <td> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Orthocenter_proof_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Orthocenter_proof_el.svg/150px-Orthocenter_proof_el.svg.png" decoding="async" width="150" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Orthocenter_proof_el.svg/225px-Orthocenter_proof_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Orthocenter_proof_el.svg/300px-Orthocenter_proof_el.svg.png 2x" data-file-width="178" data-file-height="183" /></a><figcaption>Σχήμα απόδειξης για το ορθόκεντρο.</figcaption></figure> <p><span id="Απόδειξη_με_εγγεγραμμένα_τετράπλευρα"></span> Έστω τρίγωνο <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span>. Τα ύψη <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 {\rm {BE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328e8592a6bf7d8435825af3db76b35ec778c976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.228ex; height:2.176ex;" alt="{\displaystyle {\rm {BE}}}"></span> και <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 {\rm {\Gamma Z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">Z</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma Z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73137c5353f852d419b5901a93a3eca84b083cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.873ex; height:2.176ex;" alt="{\displaystyle {\rm {\Gamma Z}}}"></span> τέμνονται στο <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 {\rm {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d5d424a20363b5b429c22bf0fa9b5eb612c429" 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 {\rm {H}}}"></span>. Θα αποδείξουμε ότι και η <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 {\rm {AH}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AH}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f77898b92ad59bf56e1e3a5d0a3fdaefeb9229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.486ex; height:2.176ex;" alt="{\displaystyle {\rm {AH}}}"></span> είναι κάθετη στην <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 {\rm {B\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376c40198ce1f0b8abb3582720709e7bfb986d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\rm {B\Gamma }}}"></span>, δλδ ύψος του τριγώνου <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span>. </p> <ul><li>Το τετράπλευρο <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 {\rm {BZE\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BZE\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aae9224629d8fff19bb6f95ab87c0e5a3130452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.101ex; height:2.176ex;" alt="{\displaystyle {\rm {BZE\Gamma }}}"></span> είναι <a href="/wiki/%CE%95%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF_%CF%84%CE%B5%CF%84%CF%81%CE%AC%CF%80%CE%BB%CE%B5%CF%85%CF%81%CE%BF" title="Εγγεγραμμένο τετράπλευρο">εγγράψιμο</a>, διότι έχει τις γωνίες <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 \angle {\rm {BZ\Gamma }}=\angle {\rm {\Gamma EB}}=90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {BZ\Gamma }}=\angle {\rm {\Gamma EB}}=90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/566786d17bf2eb991f347c7c96699073f6dabf21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.132ex; height:2.343ex;" alt="{\displaystyle \angle {\rm {BZ\Gamma }}=\angle {\rm {\Gamma EB}}=90^{\circ }}"></span>. Άρα και οι γωνίες <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 \angle {\rm {ZEB}}=\angle {\rm {Z\Gamma B}}=\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>=</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {ZEB}}=\angle {\rm {Z\Gamma B}}=\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206a8fcb651d3116f9987550a9f6b53bf5d4da40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.166ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {ZEB}}=\angle {\rm {Z\Gamma B}}=\omega }"></span> (βαίνουν στο ίδιο τόξο).</li> <li>Το τετράπλευρο <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 {\rm {AZHE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AZHE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653f66714b2b272a91920ed4cd0eda7cf29511bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.489ex; height:2.176ex;" alt="{\displaystyle {\rm {AZHE}}}"></span> είναι εγγράψιμο διότι έχει τις απέναντι γωνίες του <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 \angle {\rm {AEH}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {AEH}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90679490cc49a407532b18c5d82247c1b60d770" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.747ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {AEH}}}"></span> και <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 \angle {\rm {AZH}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {AZH}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816875d61f87f7d1bb984422b062c95ce3ade3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {AZH}}}"></span> <a href="/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AD%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B5%CF%82" title="Παραπληρωματικές γωνίες">παραπληρωματικές</a> (<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 \angle {\rm {AEH+\angle {\rm {AZH=90^{\circ }+90^{\circ }=180^{\circ }}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">H</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">H</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {AEH+\angle {\rm {AZH=90^{\circ }+90^{\circ }=180^{\circ }}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d5f05dd7812c8b1cab2d00d97c381fa2096121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.509ex; height:2.509ex;" alt="{\displaystyle \angle {\rm {AEH+\angle {\rm {AZH=90^{\circ }+90^{\circ }=180^{\circ }}}}}}"></span>). Άρα και οι γωνίες <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 \angle {\rm {ZAH}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {ZAH}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec978fed720caa5b62033dccb4c8c246878d769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {ZAH}}}"></span> και <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 \angle {\rm {ZEH=\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">H</mi> <mo>=</mo> <mi>ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {ZEH=\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff1f2821dae54ce13b0095be3eccecffaf97c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.969ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {ZEH=\omega }}}"></span> είναι ίσες.</li> <li>Οι γωνίες <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 \angle {\rm {AHZ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Z</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {AHZ}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b082cb918c68602aaab98bed54952ec6ed597fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {AHZ}}}"></span> και <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 \angle {\rm {\Delta H\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {\Delta H\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbb8868763af33b6f01a475b3e8da2893c32d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.81ex; height:2.176ex;" alt="{\displaystyle \angle {\rm {\Delta H\Gamma }}}"></span> είναι ίσες ως <a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B1%CE%BA%CE%BF%CF%81%CF%85%CF%86%CE%AE%CE%BD_%CE%B3%CF%89%CE%BD%CE%AF%CE%B5%CF%82" title="Κατακορυφήν γωνίες">κατακορυφήν</a>, δλδ <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 \angle {\rm {AHZ}}=\angle {\rm {\Delta H\Gamma }}=\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Z</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>=</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {AHZ}}=\angle {\rm {\Delta H\Gamma }}=\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726a2b805147bacad977a9ef94492d6c2a3dd3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.111ex; height:2.676ex;" alt="{\displaystyle \angle {\rm {AHZ}}=\angle {\rm {\Delta H\Gamma }}=\varphi }"></span>. Από το τρίγωνο <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 {\rm {AHZ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Z</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AHZ}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1afe090d117849907900a30cb847ebeca70bf9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.176ex;" alt="{\displaystyle {\rm {AHZ}}}"></span>, έχουμε ότι <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 \varphi +\omega =90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>+</mo> <mi>ω</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi +\omega =90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55460595dd8aca399a2bb87c6867fe76214f2033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.284ex; height:2.843ex;" alt="{\displaystyle \varphi +\omega =90^{\circ }}"></span>, τότε στο τρίγωνο <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 {\rm {H\Delta \Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {H\Delta \Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9ccca5ffe1720eb4417fe11565cb7481122960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.132ex; height:2.176ex;" alt="{\displaystyle {\rm {H\Delta \Gamma }}}"></span> είναι <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 \angle {\rm {A\Delta \Gamma }}=90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {\rm {A\Delta \Gamma }}=90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c18c3b9d0e3a6d4df339ed2ce18885f98f0a43e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.287ex; height:2.343ex;" alt="{\displaystyle \angle {\rm {A\Delta \Gamma }}=90^{\circ }}"></span>.</li></ul> <p>Συνεπώς τα τρία ύψη διέρχονται από το ίδιο σημείο. </p> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10213552"> <table role="presentation" class="math_proof {{ safesubst:p{{ safesubst:#iftrue: mw-collapsible mw-collapsed |1|2}}| |}}" style="display: block"> <tbody><tr> <td><strong>Απόδειξη (με διανύσματα)</strong>   </td></tr> <tr> <td> <p>Ξανά θεωρούμε <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 \mathrm {H} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3443729cecda76294d6b0df60ed2d60f9f9e2f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle \mathrm {H} '}"></span> την τομή των υψών <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 \mathrm {BH_{B}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {BH_{B}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537ee6194b92fea904020d0b87b88013a61b29ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.784ex; height:2.509ex;" alt="{\displaystyle \mathrm {BH_{B}} }"></span> και <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 \mathrm {\Gamma H_{\Gamma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {\Gamma H_{\Gamma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b14e7495b0a89fb5b177145bc64891d2631472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.455ex; height:2.509ex;" alt="{\displaystyle \mathrm {\Gamma H_{\Gamma }} }"></span>. Θεωρούμε τα διανύσματα <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 a,b,c,h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1618458fe7b20e04f6efc4561cf7c40e3c762d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.675ex; height:2.509ex;" alt="{\displaystyle a,b,c,h}"></span> των σημείων <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 \mathrm {A} ,\mathrm {B} ,\mathrm {\Gamma } ,\mathrm {H} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} ,\mathrm {B} ,\mathrm {\Gamma } ,\mathrm {H} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9f11aafc3cee7e20d6e81ea4e29b6217a6d865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.371ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} ,\mathrm {B} ,\mathrm {\Gamma } ,\mathrm {H} '}"></span> αντίστοιχα. Αφού <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 \mathrm {BH_{B}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {BH_{B}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537ee6194b92fea904020d0b87b88013a61b29ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.784ex; height:2.509ex;" alt="{\displaystyle \mathrm {BH_{B}} }"></span> είναι ύψος είναι κάθετο στο <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 {\rm {A\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319ed750ea4cf213d690a02f8f7af28709d80d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.196ex; height:2.176ex;" alt="{\displaystyle {\rm {A\Gamma }}}"></span> και έχουμε ότι </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 {\vec {\rm {BH'}}}\cdot {\vec {\rm {A\Gamma }}}=0\Rightarrow (h-b)\cdot (c-a)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msup> <mi mathvariant="normal">H</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\rm {BH'}}}\cdot {\vec {\rm {A\Gamma }}}=0\Rightarrow (h-b)\cdot (c-a)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3733f88ec159b3f900c7e610581fffc51d225331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.636ex; height:4.509ex;" alt="{\displaystyle {\vec {\rm {BH'}}}\cdot {\vec {\rm {A\Gamma }}}=0\Rightarrow (h-b)\cdot (c-a)=0}"></span>.</dd></dl> <p>Αφού <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 \mathrm {\Gamma H_{\Gamma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {\Gamma H_{\Gamma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b14e7495b0a89fb5b177145bc64891d2631472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.455ex; height:2.509ex;" alt="{\displaystyle \mathrm {\Gamma H_{\Gamma }} }"></span> είναι ύψος έχουμε ότι </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 {\vec {\rm {\Gamma H'}}}\cdot {\vec {\rm {AB}}}=0\Rightarrow (h-c)\cdot (b-a)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msup> <mi mathvariant="normal">H</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\rm {\Gamma H'}}}\cdot {\vec {\rm {AB}}}=0\Rightarrow (h-c)\cdot (b-a)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc6e93598ee08657b8d677111523d6dfecb5621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.636ex; height:4.509ex;" alt="{\displaystyle {\vec {\rm {\Gamma H'}}}\cdot {\vec {\rm {AB}}}=0\Rightarrow (h-c)\cdot (b-a)=0}"></span>.</dd></dl> <p>Αφαιρώντας τις δύο σχέσεις κατά μέλη λαμβάνουμε ότι </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 (h\cdot c-b\cdot c-a\cdot h+a\cdot b)-(h\cdot b-c\cdot b-a\cdot h+a\cdot c)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>⋅</mo> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo>⋅</mo> <mi>h</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>⋅</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo>⋅</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>⋅</mo> <mi>h</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\cdot c-b\cdot c-a\cdot h+a\cdot b)-(h\cdot b-c\cdot b-a\cdot h+a\cdot c)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de0ce61c5d6bd8b8ed883dbbc70450d17b841b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.488ex; height:2.843ex;" alt="{\displaystyle (h\cdot c-b\cdot c-a\cdot h+a\cdot b)-(h\cdot b-c\cdot b-a\cdot h+a\cdot c)=0}"></span></dd></dl> <p>Παραγοντοποιώντας τους όρους στο αριστερό μέλος, έχουμε ότι </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 (c-b)\cdot (h-a)=0\Rightarrow {\vec {\rm {B\Gamma }}}\cdot {\vec {\rm {AH'}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msup> <mi mathvariant="normal">H</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c-b)\cdot (h-a)=0\Rightarrow {\vec {\rm {B\Gamma }}}\cdot {\vec {\rm {AH'}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad683998c4a41604430b2e35533aea31a2cbc61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.636ex; height:4.509ex;" alt="{\displaystyle (c-b)\cdot (h-a)=0\Rightarrow {\vec {\rm {B\Gamma }}}\cdot {\vec {\rm {AH'}}}=0}"></span>,</dd></dl> <p>και άρα το <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 \mathrm {AH'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msup> <mi mathvariant="normal">H</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73a49b0d2c1fc29ac6d3c5ede56d2f9b46458bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.171ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH'} }"></span> είναι επίσης ύψος. Συνεπώς τα τρία ύψη διέρχονται από το ίδιο σημείο, το <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 \mathrm {H} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3443729cecda76294d6b0df60ed2d60f9f9e2f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle \mathrm {H} '}"></span>. </p> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10213552"> <table role="presentation" class="math_proof {{ safesubst:p{{ safesubst:#iftrue: mw-collapsible mw-collapsed |1|2}}| |}}" style="display: block"> <tbody><tr> <td><strong>Απόδειξη (με θεώρημα του Τσέβα)</strong>   </td></tr> <tr> <td> <p>Τα τρίγωνα <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 \mathrm {AH_{B}B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH_{B}B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8af8fa38915c50068bd4743cb6f6f2edb45798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH_{B}B} }"></span> και <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 \mathrm {AH_{\Gamma }\Gamma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mi mathvariant="normal">Γ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH_{\Gamma }\Gamma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03da9028a009da1bd33cb175a841bd97ec3f2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.199ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH_{\Gamma }\Gamma } }"></span> είναι όμοια καθώς έχουν μία ορθή και την <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 {\hat {\mathrm {A} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathrm {A} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78be14c8b53c3485b9f9e2053573d3a7ae10f08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathrm {A} }}}"></span> ίση. Επομένως, </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 {\frac {\mathrm {AB} }{\mathrm {A\Gamma } }}={\frac {\mathrm {AH_{B}} }{\mathrm {AH_{\Gamma }} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {AB} }{\mathrm {A\Gamma } }}={\frac {\mathrm {AH_{B}} }{\mathrm {AH_{\Gamma }} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5968c90d859fc40fb66f7abaa607b61451bf118b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.041ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {AB} }{\mathrm {A\Gamma } }}={\frac {\mathrm {AH_{B}} }{\mathrm {AH_{\Gamma }} }}}"></span>.</dd></dl> <p>Αντίστοιχα, </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 {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}={\frac {\mathrm {BH_{A}} }{\mathrm {BH_{\Gamma }} }}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}={\frac {\mathrm {BH_{A}} }{\mathrm {BH_{\Gamma }} }}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c658404b389416373e885740da001f7604ce58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.335ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}={\frac {\mathrm {BH_{A}} }{\mathrm {BH_{\Gamma }} }}\quad }"></span> και <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 \quad {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}={\frac {\mathrm {\Gamma H_{B}} }{\mathrm {\Gamma H_{A}} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}={\frac {\mathrm {\Gamma H_{B}} }{\mathrm {\Gamma H_{A}} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70ca0656e7a2297f15ae7adcf6939d3d458596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.95ex; height:5.676ex;" alt="{\displaystyle \quad {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}={\frac {\mathrm {\Gamma H_{B}} }{\mathrm {\Gamma H_{A}} }}}"></span>.</dd></dl> <p>Επομένως, </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 {\frac {\mathrm {AH_{B}} }{\mathrm {H_{B}\Gamma } }}\cdot {\frac {\mathrm {BH_{\Gamma }} }{\mathrm {H_{\Gamma }A} }}\cdot {\frac {\mathrm {\Gamma H_{A}} }{\mathrm {H_{A}B} }}={\frac {\mathrm {A\Gamma } }{\mathrm {AB} }}\cdot {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}\cdot {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mi mathvariant="normal">A</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mi mathvariant="normal">B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {AH_{B}} }{\mathrm {H_{B}\Gamma } }}\cdot {\frac {\mathrm {BH_{\Gamma }} }{\mathrm {H_{\Gamma }A} }}\cdot {\frac {\mathrm {\Gamma H_{A}} }{\mathrm {H_{A}B} }}={\frac {\mathrm {A\Gamma } }{\mathrm {AB} }}\cdot {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}\cdot {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5df0199c44f11ea537883d07be64e7f26d39f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.547ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {AH_{B}} }{\mathrm {H_{B}\Gamma } }}\cdot {\frac {\mathrm {BH_{\Gamma }} }{\mathrm {H_{\Gamma }A} }}\cdot {\frac {\mathrm {\Gamma H_{A}} }{\mathrm {H_{A}B} }}={\frac {\mathrm {A\Gamma } }{\mathrm {AB} }}\cdot {\frac {\mathrm {BA} }{\mathrm {B\Gamma } }}\cdot {\frac {\mathrm {\Gamma B} }{\mathrm {\Gamma A} }}=1}"></span>,</dd></dl> <p>και από το <a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CE%A4%CF%83%CE%AD%CE%B2%CE%B1" class="mw-redirect" title="Θεώρημα του Τσέβα">αντίστροφο θεώρημα του Τσέβα</a>, προκύπτει ότι τα τρία ύψη συντρέχουν. </p> </td></tr></tbody></table> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Ιδιότητες"><span id=".CE.99.CE.B4.CE.B9.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82"></span>Ιδιότητες</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=2" title="Επεξεργασία ενότητας: Ιδιότητες" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=2" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ιδιότητες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Σε ένα τρίγωνο <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span> με ύψη <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 {\rm {AH_{\rm {A}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AH_{\rm {A}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78df0244748a9a1403db1bd250e58c082cc6428c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.951ex; height:2.509ex;" alt="{\displaystyle {\rm {AH_{\rm {A}}}}}"></span>, <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 {\rm {BH_{\rm {B}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BH_{\rm {B}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6858512772aa20816bf394b50fe15e3f5dbf0e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.784ex; height:2.509ex;" alt="{\displaystyle {\rm {BH_{\rm {B}}}}}"></span> και <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 {\rm {\Gamma H_{\rm {\Gamma }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma H_{\rm {\Gamma }}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73eb20426d09f69965a5cef2fdb7477f793734e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.455ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma H_{\rm {\Gamma }}}}}"></span> και ορθόκεντρο <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 {\rm {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d5d424a20363b5b429c22bf0fa9b5eb612c429" 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 {\rm {H}}}"></span>, ισχύουν οι εξής ιδιότητες: </p> <ul><li>(<b><a href="/wiki/%CE%95%CF%85%CE%B8%CE%B5%CE%AF%CE%B1_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" class="mw-redirect" title="Ευθεία του Όιλερ">Ευθεία του Όιλερ</a></b>) Το <a href="/wiki/%CE%92%CE%B1%CF%81%CF%8D%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Βαρύκεντρο τριγώνου">βαρύκεντρο</a> <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 \mathrm {G} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {G} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d895cb984f1cafde4d7c7f4993e6b92d72b6ad15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.824ex; height:2.176ex;" alt="{\displaystyle \mathrm {G} }"></span>, το ορθόκεντρο <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 \mathrm {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32db8e791eaa12e32afc8fc1d60386643e43e315" 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 \mathrm {H} }"></span> και το περίκεντρο <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 \mathrm {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d6d4173d32feed308e80dbaf00e1274f40702d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathrm {O} }"></span> είναι συγγραμμικά και <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 \mathrm {HG} =2\cdot \mathrm {GO} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">G</mi> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {HG} =2\cdot \mathrm {GO} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83ade0967f5c18046aa2f3d7a4b15618ac4f3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.14ex; height:2.176ex;" alt="{\displaystyle \mathrm {HG} =2\cdot \mathrm {GO} }"></span>.</li> <li>(<b><a href="/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Κύκλος του Όιλερ">Κύκλος του Όιλερ</a></b>) Το σημεία <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 \mathrm {H_{\mathrm {A} }} ,\mathrm {H_{\mathrm {B} }} ,\mathrm {H_{\mathrm {\Gamma } }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H_{\mathrm {A} }} ,\mathrm {H_{\mathrm {B} }} ,\mathrm {H_{\mathrm {\Gamma } }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c842392f55a07d6b6816a58c04eade3fb30aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.417ex; height:2.509ex;" alt="{\displaystyle \mathrm {H_{\mathrm {A} }} ,\mathrm {H_{\mathrm {B} }} ,\mathrm {H_{\mathrm {\Gamma } }} }"></span>, τα μέσα των <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 \mathrm {AH} ,\mathrm {BH} ,\mathrm {\Gamma H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH} ,\mathrm {BH} ,\mathrm {\Gamma H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bab92e95cbec0d2dbb6ffba01394e6e2c8af2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.139ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH} ,\mathrm {BH} ,\mathrm {\Gamma H} }"></span> και τα μέσα των πλευρών ανήκουν στον ίδιο κύκλο.</li> <li>Το συμμετρικό σημείο του ορθόκεντρου ως προς κάθε μία από τις πλευρές είναι σημείο του περιγεγραμμένου κύκλου.<sup id="cite_ref-Tav_1-1" class="reference"><a href="#cite_note-Tav-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 77">: 77 </span></sup><sup id="cite_ref-T57_2-1" class="reference"><a href="#cite_note-T57-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 270">: 270 </span></sup></li> <li>Το συμμετρικό σημείο του ορθόκεντρου ως προς το μέσο κάθε μίας από τις πλευρές του είναι σημείο του περιγεγραμμένου κύκλου.<sup id="cite_ref-Tav_1-2" class="reference"><a href="#cite_note-Tav-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 76">: 76 </span></sup></li> <li>Το ορθόκεντρο είναι το σημείο <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 \mathrm {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72172888980d0d3565baec875a4c3e8eed50ed26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {P} }"></span> που ελαχιστοποιεί την ακόλουθη <a href="/wiki/%CE%A3%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Συνάρτηση">συνάρτηση</a>:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><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 f(\mathrm {P} )=\mathrm {PA} +\mathrm {PB} +\mathrm {P\Gamma } +\mathrm {PH_{A}} +\mathrm {PH_{B}} +\mathrm {PH_{\Gamma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">B</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathrm {P} )=\mathrm {PA} +\mathrm {PB} +\mathrm {P\Gamma } +\mathrm {PH_{A}} +\mathrm {PH_{B}} +\mathrm {PH_{\Gamma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd040fedd2bb7c02e888ef6c9753692b23270f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.659ex; height:2.843ex;" alt="{\displaystyle f(\mathrm {P} )=\mathrm {PA} +\mathrm {PB} +\mathrm {P\Gamma } +\mathrm {PH_{A}} +\mathrm {PH_{B}} +\mathrm {PH_{\Gamma }} }"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Μετρικές_σχέσεις"><span id=".CE.9C.CE.B5.CF.84.CF.81.CE.B9.CE.BA.CE.AD.CF.82_.CF.83.CF.87.CE.AD.CF.83.CE.B5.CE.B9.CF.82"></span>Μετρικές σχέσεις</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=3" title="Επεξεργασία ενότητας: Μετρικές σχέσεις" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=3" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Μετρικές σχέσεις"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Έστω <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 \mathrm {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1811407dea8b43727d28dbe8da7251985b03e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {E} }"></span> το <a href="/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C" class="mw-redirect" title="Εμβαδό">εμβαδό</a> του τριγώνου και <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 {\hat {\mathrm {A} }}>90^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>></mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathrm {A} }}>90^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c72a64ca28e665f62928d9fda103c41ca56fc9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.196ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathrm {A} }}>90^{o}}"></span>, τότε<sup id="cite_ref-P74_4-1" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 47">: 47 </span></sup></li></ul> <dl><dd><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 \mathrm {AH} ^{2}={\frac {\alpha \cdot (\beta ^{2}+\gamma ^{2}-\alpha ^{2})}{4\mathrm {E} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH} ^{2}={\frac {\alpha \cdot (\beta ^{2}+\gamma ^{2}-\alpha ^{2})}{4\mathrm {E} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d46d0e5c748125b314cef7ba61a88fe8abe0578" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.399ex; height:5.843ex;" alt="{\displaystyle \mathrm {AH} ^{2}={\frac {\alpha \cdot (\beta ^{2}+\gamma ^{2}-\alpha ^{2})}{4\mathrm {E} }}}"></span>,</dd></dl></dd> <dd>και αν <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 \mathrm {A} <90^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} <90^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959758d0b304d61f814bec0e2a14d12b3851919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.196ex; height:2.343ex;" alt="{\displaystyle \mathrm {A} <90^{o}}"></span>, τότε <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 \mathrm {AH} ^{2}={\frac {\alpha \cdot (\alpha ^{2}-\beta ^{2}-\gamma ^{2})}{4\mathrm {E} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH} ^{2}={\frac {\alpha \cdot (\alpha ^{2}-\beta ^{2}-\gamma ^{2})}{4\mathrm {E} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a8280834004a0ac295f75a04dd9d8217e9e4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.399ex; height:5.843ex;" alt="{\displaystyle \mathrm {AH} ^{2}={\frac {\alpha \cdot (\alpha ^{2}-\beta ^{2}-\gamma ^{2})}{4\mathrm {E} }}}"></span>.</dd></dl></dd></dl> <ul><li>Αν <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 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></span><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}"></span> η ακτίνα του <a href="/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" class="mw-redirect" title="Περιγεγραμμένος κύκλος">περιγεγραμμένου κύκλου</a> του τριγώνου, τότε</li></ul> <dl><dd><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 \mathrm {AH} ^{2}+\mathrm {BH} ^{2}+\mathrm {\Gamma H} ^{2}=12R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>12</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH} ^{2}+\mathrm {BH} ^{2}+\mathrm {\Gamma H} ^{2}=12R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035c1035c79823a80e90f12a2cc539dc78189378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.754ex; height:3.176ex;" alt="{\displaystyle \mathrm {AH} ^{2}+\mathrm {BH} ^{2}+\mathrm {\Gamma H} ^{2}=12R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"></span></dd></dl></dd></dl> <ul><li>Αν <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 \mathrm {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32db8e791eaa12e32afc8fc1d60386643e43e315" 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 \mathrm {H} }"></span> το ορθόκεντρο, <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 \mathrm {G} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {G} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d895cb984f1cafde4d7c7f4993e6b92d72b6ad15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.824ex; height:2.176ex;" alt="{\displaystyle \mathrm {G} }"></span> το <a href="/wiki/%CE%92%CE%B1%CF%81%CF%8D%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Βαρύκεντρο τριγώνου">βαρύκεντρο</a>, <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 (\mathrm {O} ,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {O} ,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8739026455cdb14e11c3bc802364ec5db678440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (\mathrm {O} ,R)}"></span> ο <a href="/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" class="mw-redirect" title="Περιγεγραμμένος κύκλος">περιγεγραμμένος</a>, <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 (\mathrm {I} ,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {I} ,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680b46f5e665e048a82d54016caf05a4ef034f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.885ex; height:2.843ex;" alt="{\displaystyle (\mathrm {I} ,\rho )}"></span> o <a href="/wiki/%CE%95%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Εγγεγραμμένος κύκλος τριγώνου">εγγεγραμμένος</a> και <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 (\mathrm {I_{A}} ,\rho _{\mathrm {A} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {I_{A}} ,\rho _{\mathrm {A} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cd34b15bd4032ca64e212600dc786a22fbe297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.814ex; height:2.843ex;" alt="{\displaystyle (\mathrm {I_{A}} ,\rho _{\mathrm {A} })}"></span> ο <a href="/wiki/%CE%A0%CE%B1%CF%81%CE%B5%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CE%B9_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CE%B9_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Παρεγγεγραμμένοι κύκλοι τριγώνου">παρεγγεγραμμένος</a> κύκλος, τότε<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-P74_4-2" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 47">: 47 </span></sup></li></ul> <dl><dd><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 \mathrm {OH} ^{2}=9R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>9</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {OH} ^{2}=9R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0b158aea3eaf3e6a7b9507dcf34237b4384d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.282ex; height:3.176ex;" alt="{\displaystyle \mathrm {OH} ^{2}=9R^{2}-(\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"></span>,</dd> <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 \mathrm {HG} ^{2}=4R^{2}-4\cdot (\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {HG} ^{2}=4R^{2}-4\cdot (\alpha ^{2}+\beta ^{2}+\gamma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a51d1a6350673aa82bd9ec013f55149760b59073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.14ex; height:3.176ex;" alt="{\displaystyle \mathrm {HG} ^{2}=4R^{2}-4\cdot (\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"></span>,</dd> <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 \mathrm {HI} ^{2}=2\rho ^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {HI} ^{2}=2\rho ^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3886e8aa65214c08fafdd94ea5a7332df32b28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.473ex; height:3.176ex;" alt="{\displaystyle \mathrm {HI} ^{2}=2\rho ^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }"></span>,</dd> <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 \mathrm {HI_{A}} ^{2}=2\rho _{\mathrm {A} }^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {HI_{A}} ^{2}=2\rho _{\mathrm {A} }^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92814d297304bc17e687e1a5896f28c7aaa6a08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.348ex; height:3.343ex;" alt="{\displaystyle \mathrm {HI_{A}} ^{2}=2\rho _{\mathrm {A} }^{2}-4R^{2}\cos \mathrm {A} \cos \mathrm {B} \cos \mathrm {\Gamma } }"></span>.</dd></dl></dd></dl> <ul><li>Οι τριγραμμικές συντεταγμένες του ορθόκεντρου είναι</li></ul> <dl><dd><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 \sec A:\sec B:\sec C=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>A</mi> <mo>:</mo> <mi>sec</mi> <mo>⁡</mo> <mi>B</mi> <mo>:</mo> <mi>sec</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>A</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> <mo>:</mo> <mi>cos</mi> <mo>⁡</mo> <mi>B</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mo>:</mo> <mi>cos</mi> <mo>⁡</mo> <mi>C</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec A:\sec B:\sec C=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ccbcef832c48c845da1377b6a35fcad8b8761d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:81.679ex; height:2.343ex;" alt="{\displaystyle \sec A:\sec B:\sec C=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B}"></span>,</dd></dl></dd> <dd>και οι βαρυκεντρικές του συντεταγμένες είναι <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 \tan A:\tan B:\tan C=(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(\alpha ^{2}-\beta ^{2}+\gamma ^{2}):(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2}):(\alpha ^{2}-\beta ^{2}+\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <mi>B</mi> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan A:\tan B:\tan C=(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(\alpha ^{2}-\beta ^{2}+\gamma ^{2}):(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2}):(\alpha ^{2}-\beta ^{2}+\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e012df72425e5ec4e42ff9f89e9a6112d8761a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:119.521ex; height:3.176ex;" alt="{\displaystyle \tan A:\tan B:\tan C=(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(\alpha ^{2}-\beta ^{2}+\gamma ^{2}):(\alpha ^{2}+\beta ^{2}-\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2}):(\alpha ^{2}-\beta ^{2}+\gamma ^{2})(-\alpha ^{2}+\beta ^{2}+\gamma ^{2})}"></span>.</dd></dl></dd></dl> <ul><li>Για τα ύψη <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 \mathrm {AH_{A}} ,\mathrm {BH_{B}} ,\mathrm {\Gamma H_{\Gamma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH_{A}} ,\mathrm {BH_{B}} ,\mathrm {\Gamma H_{\Gamma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae119d7d6bf9ca9f736ad6e3d7a9c8d5c0b9cf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.259ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH_{A}} ,\mathrm {BH_{B}} ,\mathrm {\Gamma H_{\Gamma }} }"></span>, ισχύει ότι <ul><li><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 \mathrm {AH} \cdot \mathrm {HH_{A}} =\mathrm {BH} \cdot \mathrm {HH_{B}} =\mathrm {\Gamma H} \cdot \mathrm {HH_{\Gamma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AH} \cdot \mathrm {HH_{A}} =\mathrm {BH} \cdot \mathrm {HH_{B}} =\mathrm {\Gamma H} \cdot \mathrm {HH_{\Gamma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ffc844e5522d1ad3622fcd139f3c902bcdba9e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.884ex; height:2.509ex;" alt="{\displaystyle \mathrm {AH} \cdot \mathrm {HH_{A}} =\mathrm {BH} \cdot \mathrm {HH_{B}} =\mathrm {\Gamma H} \cdot \mathrm {HH_{\Gamma }} }"></span>,</li> <li><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 {\frac {\mathrm {AH} }{\mathrm {HH_{A}} }}+{\frac {\mathrm {BH} }{\mathrm {HH_{B}} }}+{\frac {\mathrm {\Gamma H} }{\mathrm {HH_{\Gamma }} }}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {AH} }{\mathrm {HH_{A}} }}+{\frac {\mathrm {BH} }{\mathrm {HH_{B}} }}+{\frac {\mathrm {\Gamma H} }{\mathrm {HH_{\Gamma }} }}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28830a97b6b17b848475c74ce1a2509e6f38932e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.029ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {AH} }{\mathrm {HH_{A}} }}+{\frac {\mathrm {BH} }{\mathrm {HH_{B}} }}+{\frac {\mathrm {\Gamma H} }{\mathrm {HH_{\Gamma }} }}=1}"></span>.</li> <li><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 (\upsilon _{\mathrm {A} }+\upsilon _{\mathrm {B} }+\upsilon _{\mathrm {\Gamma } })\cdot \left({\frac {1}{\upsilon _{\mathrm {A} }}}+{\frac {1}{\upsilon _{\mathrm {B} }}}+{\frac {1}{\upsilon _{\mathrm {\Gamma } }}}\right)=(\alpha +\beta +\gamma )\cdot \left({\frac {1}{\alpha }}+{\frac {1}{\beta }}+{\frac {1}{\gamma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>υ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\upsilon _{\mathrm {A} }+\upsilon _{\mathrm {B} }+\upsilon _{\mathrm {\Gamma } })\cdot \left({\frac {1}{\upsilon _{\mathrm {A} }}}+{\frac {1}{\upsilon _{\mathrm {B} }}}+{\frac {1}{\upsilon _{\mathrm {\Gamma } }}}\right)=(\alpha +\beta +\gamma )\cdot \left({\frac {1}{\alpha }}+{\frac {1}{\beta }}+{\frac {1}{\gamma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4394cc76128435907da49bb55e30801bdbcfa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:68.594ex; height:6.176ex;" alt="{\displaystyle (\upsilon _{\mathrm {A} }+\upsilon _{\mathrm {B} }+\upsilon _{\mathrm {\Gamma } })\cdot \left({\frac {1}{\upsilon _{\mathrm {A} }}}+{\frac {1}{\upsilon _{\mathrm {B} }}}+{\frac {1}{\upsilon _{\mathrm {\Gamma } }}}\right)=(\alpha +\beta +\gamma )\cdot \left({\frac {1}{\alpha }}+{\frac {1}{\beta }}+{\frac {1}{\gamma }}\right)}"></span>.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Δείτε_επίσης"><span id=".CE.94.CE.B5.CE.AF.CF.84.CE.B5_.CE.B5.CF.80.CE.AF.CF.83.CE.B7.CF.82"></span>Δείτε επίσης</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=4" title="Επεξεργασία ενότητας: Δείτε επίσης" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=4" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Δείτε επίσης"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%CE%8E%CF%88%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Ύψος τριγώνου">Ύψος τριγώνου</a></li> <li><a href="/wiki/%CE%92%CE%B1%CF%81%CF%8D%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Βαρύκεντρο τριγώνου">Βαρύκεντρο τριγώνου</a>, το σημείο τομής των <a href="/wiki/%CE%94%CE%B9%CE%AC%CE%BC%CE%B5%CF%83%CE%BF%CF%82_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Διάμεσος (γεωμετρία)">διαμέσων</a> του τριγώνου</li> <li><a href="/wiki/%CE%88%CE%B3%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Έγκεντρο τριγώνου">Έγκεντρο τριγώνου</a>, το σημείο τομής των <a href="/wiki/%CE%94%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1%CF%82" title="Διχοτόμος γωνίας">διχοτόμων</a> του τριγώνου</li> <li><a href="/wiki/%CE%A0%CE%B5%CF%81%CE%AF%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Περίκεντρο τριγώνου">Περίκεντρο τριγώνου</a>, το σημείο τομής των <a href="/wiki/%CE%9C%CE%B5%CF%83%CE%BF%CE%BA%CE%AC%CE%B8%CE%B5%CF%84%CE%BF%CF%82" class="mw-redirect" title="Μεσοκάθετος">μεσοκαθέτων</a> του τριγώνου</li></ul> <div class="mw-heading mw-heading2"><h2 id="Σημειώσεις"><span id=".CE.A3.CE.B7.CE.BC.CE.B5.CE.B9.CF.8E.CF.83.CE.B5.CE.B9.CF.82"></span>Σημειώσεις</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=5" title="Επεξεργασία ενότητας: Σημειώσεις" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=5" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Σημειώσεις"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Οι ευθείες αυτές τέμνονται ανά δύο, καθώς είναι παράλληλες στα ευθύγραμμα τμήματα του τριγώνου <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span> που τέμονται στις κορυφές του τριγώνου.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Το τρίγωνο αυτό λέγεται το <a href="/wiki/%CE%91%CE%BD%CF%84%CE%B9%CF%83%CF%85%CE%BC%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" class="mw-redirect" title="Αντισυμπληρωματικό τρίγωνο">αντισυμπληρωματικό</a> του τριγώνου <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 {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Παραπομπές"><span id=".CE.A0.CE.B1.CF.81.CE.B1.CF.80.CE.BF.CE.BC.CF.80.CE.AD.CF.82"></span>Παραπομπές</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&veaction=edit&section=6" title="Επεξεργασία ενότητας: Παραπομπές" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85&action=edit&section=6" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Παραπομπές"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Tav-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Tav_1-0">1,0</a></sup> <sup><a href="#cite_ref-Tav_1-1">1,1</a></sup> <sup><a href="#cite_ref-Tav_1-2">1,2</a></sup></span> <span class="reference-text"><cite class="citation book">Ταβανλης, Χ. <i>Επίπεδος Γεωμετρία</i>. Αθήνα: Ι. Χιωτελη.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%95%CF%80%CE%AF%CF%80%CE%B5%CE%B4%CE%BF%CF%82+%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%99.+%CE%A7%CE%B9%CF%89%CF%84%CE%B5%CE%BB%CE%B7&rft.aulast=%CE%A4%CE%B1%CE%B2%CE%B1%CE%BD%CE%BB%CE%B7%CF%82&rft.aufirst=%CE%A7.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-T57-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-T57_2-0">2,0</a></sup> <sup><a href="#cite_ref-T57_2-1">2,1</a></sup></span> <span class="reference-text"><cite class="citation book">Τόγκας, Πέτρος Γ. (1957). <i>Θεωρητική Γεωμετρία</i>. Αθήνα: Πέτρου Γ. Τόγκα.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%98%CE%B5%CF%89%CF%81%CE%B7%CF%84%CE%B9%CE%BA%CE%AE+%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%85+%CE%93.+%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1&rft.date=1957&rft.aulast=%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1%CF%82&rft.aufirst=%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%82+%CE%93.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-A75-3"><span class="mw-cite-backlink"><a href="#cite_ref-A75_3-0">↑</a></span> <span class="reference-text"><cite class="citation book">Αλεξίου, Κ. Τ. (1975). <i>Θεωρητική Γεωμετρία: Τεύχος Α'<span></span></i>. Αθήνα.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%98%CE%B5%CF%89%CF%81%CE%B7%CF%84%CE%B9%CE%BA%CE%AE+%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1%3A+%CE%A4%CE%B5%CF%8D%CF%87%CE%BF%CF%82+%CE%91%27&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.date=1975&rft.aulast=%CE%91%CE%BB%CE%B5%CE%BE%CE%AF%CE%BF%CF%85&rft.aufirst=%CE%9A.+%CE%A4.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-P74-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-P74_4-0">4,0</a></sup> <sup><a href="#cite_ref-P74_4-1">4,1</a></sup> <sup><a href="#cite_ref-P74_4-2">4,2</a></sup></span> <span class="reference-text"><cite class="citation book">Πανάκης, Ιωάννης (1974). <i>Μαθηματικά Δ',Ε',ΣΤ' Γυμνασίου Τόμος Δεύτερος</i>. Αθήνα: Οργανισμός Εκδόσεως Διδακτικών Βιβλίων.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC+%CE%94%27%2C%CE%95%27%2C%CE%A3%CE%A4%27+%CE%93%CF%85%CE%BC%CE%BD%CE%B1%CF%83%CE%AF%CE%BF%CF%85+%CE%A4%CF%8C%CE%BC%CE%BF%CF%82+%CE%94%CE%B5%CF%8D%CF%84%CE%B5%CF%81%CE%BF%CF%82&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%9F%CF%81%CE%B3%CE%B1%CE%BD%CE%B9%CF%83%CE%BC%CF%8C%CF%82+%CE%95%CE%BA%CE%B4%CF%8C%CF%83%CE%B5%CF%89%CF%82+%CE%94%CE%B9%CE%B4%CE%B1%CE%BA%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CF%89%CE%BD&rft.date=1974&rft.aulast=%CE%A0%CE%B1%CE%BD%CE%AC%CE%BA%CE%B7%CF%82&rft.aufirst=%CE%99%CF%89%CE%AC%CE%BD%CE%BD%CE%B7%CF%82&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="citation Journal">Hiriart-Urruty, Jean-Baptiste; Laurent, Pierre-Jean (2015). «A characterization by optimization of the orthocenter of a triangle». <i>Elemente der Mathematik</i> <b>70</b> (2): 45–48. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.4171%2FEM%2F273">10.4171/EM/273</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+characterization+by+optimization+of+the+orthocenter+of+a+triangle&rft.jtitle=Elemente+der+Mathematik&rft.aulast=Hiriart-Urruty&rft.aufirst=Jean-Baptiste&rft.au=Hiriart-Urruty%2C%26%2332%3BJean-Baptiste&rft.au=Laurent%2C+Pierre-Jean&rft.date=2015&rft.volume=70&rft.issue=2&rft.pages=45%E2%80%9348&rft_id=info:doi/10.4171%2FEM%2F273&rfr_id=info:sid/el.wikipedia.org:%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><cite class="citation web">Bogomolny, Alexander. <a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/arithmetic/algebra/DistanceOH.shtml">«Distance between the Orthocenter and Circumcenter»</a>. Cut the Knot<span class="reference-accessdate">. Ανακτήθηκε στις 3 Σεπτεμβρίου 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Distance+between+the+Orthocenter+and+Circumcenter&rft.pub=Cut+the+Knot&rft.aulast=Bogomolny&rft.aufirst=Alexander&rft_id=http%3A%2F%2Fwww.cut-the-knot.org%2Farithmetic%2Falgebra%2FDistanceOH.shtml&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><cite class="citation web">Yiu, Paul. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220213205615/http://math.fau.edu/yiu/AEG2013/2013AEG.pdf">«Advanced Euclidean Geometry»</a> <span style="font-size:85%;">(PDF)</span>. Department of Mathematics, Florida Atlantic University. Αρχειοθετήθηκε <a rel="nofollow" class="external text" href="http://math.fau.edu/yiu/AEG2013/2013AEG.pdf">από το πρωτότυπο</a> <span style="font-size:85%;">(PDF)</span> στις 13 Φεβρουαρίου 2022<span class="reference-accessdate">. Ανακτήθηκε στις 3 Σεπτεμβρίου 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Advanced+Euclidean+Geometry&rft.pub=Department+of+Mathematics%2C+Florida+Atlantic+University&rft.aulast=Yiu&rft.aufirst=Paul&rft_id=http%3A%2F%2Fmath.fau.edu%2Fyiu%2FAEG2013%2F2013AEG.pdf&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%9F%CF%81%CE%B8%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF+%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r10387572">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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style="width:1%">Βάσει πλευρών</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A3%CE%BA%CE%B1%CE%BB%CE%B7%CE%BD%CF%8C_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Σκαληνό τρίγωνο">σκαληνό</a></li> <li><a href="/wiki/%CE%99%CF%83%CE%BF%CF%83%CE%BA%CE%B5%CE%BB%CE%AD%CF%82_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ισοσκελές τρίγωνο">ισοσκελές</a></li> <li><a href="/wiki/%CE%99%CF%83%CF%8C%CF%80%CE%BB%CE%B5%CF%85%CF%81%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ισόπλευρο τρίγωνο">ισόπλευρο</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Άλλα</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A8%CE%B5%CF%85%CE%B4%CE%BF%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ψευδορθογώνιο τρίγωνο">ψευδορθογώνιο</a></li> <li><a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF_%CF%84%CE%BF%CF%85_%CE%89%CF%81%CF%89%CE%BD%CE%B1" title="Τρίγωνο του Ήρωνα">τρίγωνο Ήρωνα</a></li> <li><a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF_%CF%84%CE%BF%CF%85_%CE%9A%CE%AD%CF%80%CE%BB%CE%B5%CF%81" title="Τρίγωνο του Κέπλερ">τρίγωνο Κέπλερ</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Σημεία τριγώνου</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Βασικά</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%92%CE%B1%CF%81%CF%8D%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Βαρύκεντρο τριγώνου">βαρύκεντρο</a></li> <li><a href="/wiki/%CE%A0%CE%B5%CF%81%CE%AF%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Περίκεντρο τριγώνου">περίκεντρο</a></li> <li><a class="mw-selflink selflink">ορθόκεντρο</a></li> <li><a href="/wiki/%CE%88%CE%B3%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Έγκεντρο τριγώνου">έγκεντρο</a></li> <li><a href="/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%B1_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Παράκεντρα τριγώνου">παράκεντρα</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Άλλα</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A3%CE%B7%CE%BC%CE%B5%CE%AF%CE%BF_Gergonne" title="Σημείο Gergonne">σημείο Gergonne</a></li> <li><a href="/wiki/%CE%A3%CE%B7%CE%BC%CE%B5%CE%AF%CE%BF_%CE%9D%CE%AC%CE%B3%CE%BA%CE%B5%CE%BB" title="Σημείο Νάγκελ">σημείο Νάγκελ</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ευθείες τριγώνου</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%CE%A3%CE%B5%CE%B2%CE%B9%CE%B1%CE%BD%CE%AE" title="Σεβιανή">Σεβιανές</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%94%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1%CF%82#Εσωτερικές_διχοτόμοι_τριγώνου" title="Διχοτόμος γωνίας">εσωτερική διχοτόμος</a></li> <li><a href="/wiki/%CE%95%CE%BE%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%82" class="mw-redirect" title="Εξωτερική διχοτόμος">εξωτερική διχοτόμος</a></li> <li><a href="/wiki/%CE%8E%CF%88%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Ύψος τριγώνου">ύψος</a></li> <li><a href="/wiki/%CE%94%CE%B9%CE%AC%CE%BC%CE%B5%CF%83%CE%BF%CF%82_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Διάμεσος (γεωμετρία)">διάμεσος</a></li> <li><a href="/wiki/%CE%A3%CF%85%CE%BC%CE%BC%CE%B5%CF%84%CF%81%CE%BF%CE%B4%CE%B9%CE%AC%CE%BC%CE%B5%CF%83%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Συμμετροδιάμεσος τριγώνου">συμμετροδιάμεσος</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Άλλες</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%9C%CE%B5%CF%83%CE%BF%CE%BA%CE%AC%CE%B8%CE%B5%CF%84%CE%B7_%CE%B5%CF%85%CE%B8%CF%8D%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF%CF%85_%CF%84%CE%BC%CE%AE%CE%BC%CE%B1%CF%84%CE%BF%CF%82#Μεσοκάθετοι_τριγώνου" title="Μεσοκάθετη ευθύγραμμου τμήματος">μεσοκάθετοι</a></li> <li><a href="/wiki/%CE%95%CF%85%CE%B8%CE%B5%CE%AF%CE%B1_%CE%A3%CE%AF%CE%BC%CF%83%CE%BF%CE%BD-%CE%93%CE%BF%CF%85%CE%AC%CE%BB%CE%B1%CF%82" class="mw-redirect" title="Ευθεία Σίμσον-Γουάλας">ευθεία Σίμσον-Γουάλας</a></li> <li><a href="/wiki/%CE%95%CF%85%CE%B8%CE%B5%CE%AF%CE%B1_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Ευθεία Όιλερ">ευθεία Όιλερ</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Κύκλοι τριγώνου</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Βασικοί</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Περιγεγραμμένος κύκλος τριγώνου">περιγεγραμμένος</a></li> <li><a href="/wiki/%CE%95%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Εγγεγραμμένος κύκλος τριγώνου">εγγεγραμμένος</a></li> <li><a href="/wiki/%CE%A0%CE%B1%CF%81%CE%B5%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CE%B9_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CE%B9_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Παρεγγεγραμμένοι κύκλοι τριγώνου">παρεγγεγραμμένοι</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Άλλοι</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Κύκλος του Όιλερ">κύκλος του Όιλερ</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Μετρικές σχέσεις</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Αναλογίες</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%98%CE%B1%CE%BB%CE%AE" class="mw-redirect" title="Θεώρημα Θαλή">θεώρημα Θαλή</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%B4%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%85" title="Θεώρημα διχοτόμου">θεώρημα εσωτερικής και εξωτερικής διχοτόμου</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%9C%CE%B5%CE%BD%CE%B5%CE%BB%CE%AC%CE%BF%CF%85" title="Θεώρημα Μενελάου">θεώρημα Μενελάου</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%A4%CF%83%CE%AD%CE%B2%CE%B1" title="Θεώρημα Τσέβα">θεώρημα Τσέβα</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Εμβαδόν</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%89%CF%81%CF%89%CE%BD%CE%B1" title="Τύπος του Ήρωνα">τύπος του Ήρωνα</a></li> <li><a href="/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Εμβαδόν τριγώνου">εμβαδόν τριγώνου</a></li> <li><a href="/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD#Τρίγωνο" title="Εμβαδόν">λίστα τύπων για το εμβαδόν</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Μήκη σεβιανών</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A0%CF%85%CE%B8%CE%B1%CE%B3%CF%8C%CF%81%CE%B5%CE%B9%CE%BF_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Πυθαγόρειο θεώρημα">Πυθαγόρειο θεώρημα</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%A3%CF%84%CE%B9%CE%BF%CF%8D%CE%B1%CF%81%CF%84" title="Θεώρημα Στιούαρτ">θεώρημα Στιούαρτ</a></li> <li><a href="/wiki/%CE%A0%CF%81%CF%8E%CF%84%CE%BF_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%B4%CE%B9%CE%B1%CE%BC%CE%AD%CF%83%CF%89%CE%BD" title="Πρώτο θεώρημα διαμέσων">1ο</a> και <a href="/wiki/%CE%94%CE%B5%CF%8D%CF%84%CE%B5%CF%81%CE%BF_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%B4%CE%B9%CE%B1%CE%BC%CE%AD%CF%83%CF%89%CE%BD" class="mw-redirect" title="Δεύτερο θεώρημα διαμέσων">2ο</a> θεώρημα διαμέσων</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Τριγωνομετρικές<br /> σχέσεις</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CE%B7%CE%BC%CE%B9%CF%84%CF%8C%CE%BD%CF%89%CE%BD" title="Νόμος των ημιτόνων">νόμος των ημιτόνων</a></li> <li><a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CF%83%CF%85%CE%BD%CE%B7%CE%BC%CE%B9%CF%84%CF%8C%CE%BD%CF%89%CE%BD" title="Νόμος των συνημιτόνων">νόμος των συνημιτόνων</a></li> <li><a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CE%B5%CF%86%CE%B1%CF%80%CF%84%CE%BF%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Νόμος των εφαπτομένων">νόμος των εφαπτομένων</a></li> <li><a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CE%B9_Mollweide" title="Τύποι Mollweide">τύποι Mollweide</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Άλλες</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%9A%CE%B1%CF%81%CE%BD%CF%8C_(%CE%B1%CE%BA%CF%84%CE%AF%CE%BD%CE%B5%CF%82)" title="Θεώρημα Καρνό (ακτίνες)">θεώρημα Καρνό (ακτίνες)</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%9A%CE%B1%CF%81%CE%BD%CF%8C_(%CE%BA%CE%AC%CE%B8%CE%B5%CF%84%CE%BF%CE%B9)" title="Θεώρημα Καρνό (κάθετοι)">θεώρημα Καρνό (κάθετοι)</a></li> <li><a href="/wiki/%CE%A3%CF%87%CE%AD%CF%83%CE%B7_%CF%84%CE%BF%CF%85_%CE%9B%CE%AC%CE%B9%CE%BC%CF%80%CE%BD%CE%B9%CF%84%CF%82" title="Σχέση του Λάιμπνιτς">σχέση του Λάιμπνιτς</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Θεώρημα του Όιλερ (γεωμετρία)">θεώρημα Όιλερ</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Σχετικά θεωρήματα</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CF%81%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CF%89%CE%BD_%CF%84%CE%BF%CF%85_%CE%9C%CF%8C%CF%81%CE%BB%CE%B5%CF%8A" title="Θεώρημα τριχοτόμων του Μόρλεϊ">θεώρημα τριχοτόμων του Μόρλεϊ</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%9D%CE%B1%CF%80%CE%BF%CE%BB%CE%AD%CE%BF%CE%BD%CF%84%CE%B1" title="Θεώρημα Ναπολέοντα">θεώρημα Ναπολέοντα</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%A0%CE%AC%CF%80%CF%80%CE%BF%CF%85_%CE%B3%CE%B9%CE%B1_%CF%84%CE%BF_%CE%B5%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD" title="Θεώρημα Πάππου για το εμβαδόν">θεώρημα Πάππου</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_Steiner-Lehmus" title="Θεώρημα Steiner-Lehmus">θεώρημα Steiner-Lehmus</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%9D%CE%AC%CE%B3%CE%BA%CE%B5%CE%BB" title="Θεώρημα Νάγκελ">θεώρημα Νάγκελ</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%92%CE%B9%CE%B2%CE%B9%CE%AC%CE%BD%CE%B9" title="Θεώρημα Βιβιάνι">θεώρημα Βιβιάνι</a></li> <li><a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_van_Schooten" title="Θεώρημα van Schooten">θεώρημα van Schooten</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Παράγωγα τρίγωνα</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Συμπληρωματικό τρίγωνο">συμπληρωματικό</a></li> <li><a href="/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Συμπληρωματικό τρίγωνο">αντισυμπληρωματικό</a></li> <li><a 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