Εγγεγραμμένος και Παρεγγεγραμμένοι κύκλοι τριγώνου

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href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_el.wikipedia.org&uselang=el" class=""><span>Δωρεές</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" 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%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%CE%B1%CE%B9+%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" title="Σας ενθαρρύνουμε να δημιουργήσετε ένα λογαριασμό και να συνδεθείτε· ωστόσο, δεν είναι υποχρεωτικό" class=""><span>Δημιουργία λογαριασμού</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%A3%CF%8D%CE%BD%CE%B4%CE%B5%CF%83%CE%B7%CE%A7%CF%81%CE%AE%CF%83%CF%84%CE%B7&returnto=%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%CE%B1%CE%B9+%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" title="Σας ενθαρρύνουμε να συνδεθείτε· ωστόσο, δεν είναι υποχρεωτικό [o]" accesskey="o" class=""><span>Σύνδεση</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Περισσότερες επιλογές" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Προσωπικά εργαλεία" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Προσωπικά εργαλεία</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet 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 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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%A3%CF%8D%CE%BD%CE%B4%CE%B5%CF%83%CE%B7%CE%A7%CF%81%CE%AE%CF%83%CF%84%CE%B7&returnto=%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%CE%B1%CE%B9+%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" title="Σας ενθαρρύνουμε να συνδεθείτε· ωστόσο, δεν είναι υποχρεωτικό [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Σύνδεση</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Σελίδες για αποσυνδεμένους συντάκτες <a href="/wiki/%CE%92%CE%BF%CE%AE%CE%B8%CE%B5%CE%B9%CE%B1:%CE%95%CE%B9%CF%83%CE%B1%CE%B3%CF%89%CE%B3%CE%AE" aria-label="Μάθετε περισσότερα σχετικά με την επεξεργασία"><span>μάθετε περισσότερα</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%9F%CE%B9%CE%A3%CF%85%CE%BD%CE%B5%CE%B9%CF%83%CF%86%CE%BF%CF%81%CE%AD%CF%82%CE%9C%CE%BF%CF%85" title="Μια λίστα με τις επεξεργασίες που έγιναν από αυτή τη διεύθυνση IP [y]" accesskey="y"><span>Συνεισφορές</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%97%CE%A3%CF%85%CE%B6%CE%AE%CF%84%CE%B7%CF%83%CE%AE%CE%9C%CE%BF%CF%85" title="Συζήτηση σχετικά με τις αλλαγές που έγιναν από αυτή τη διεύθυνση IP [n]" accesskey="n"><span>Συζήτηση για αυτή την IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div 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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-Εγγεγραμμένος_κύκλος" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Εγγεγραμμένος_κύκλος"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Εγγεγραμμένος κύκλος</span> </div> </a> <button aria-controls="toc-Εγγεγραμμένος_κύκλος-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Εγγεγραμμένος κύκλος υποενότητας</span> </button> <ul id="toc-Εγγεγραμμένος_κύκλος-sublist" class="vector-toc-list"> <li id="toc-Αποδείξεις" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Αποδείξεις"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</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-2"> <a class="vector-toc-link" href="#Ιδιότητες"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Ιδιότητες</span> </div> </a> <ul id="toc-Ιδιότητες-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Παρεγγεγραμμένοι_κύκλοι" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Παρεγγεγραμμένοι_κύκλοι"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Παρεγγεγραμμένοι κύκλοι</span> </div> </a> <button aria-controls="toc-Παρεγγεγραμμένοι_κύκλοι-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Παρεγγεγραμμένοι κύκλοι υποενότητας</span> </button> <ul id="toc-Παρεγγεγραμμένοι_κύκλοι-sublist" class="vector-toc-list"> <li id="toc-Απόδειξη" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Απόδειξη"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Απόδειξη</span> </div> </a> <ul id="toc-Απόδειξη-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ιδιότητες_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ιδιότητες_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Ιδιότητες</span> </div> </a> <ul id="toc-Ιδιότητες_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Δείτε_επίσης" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Δείτε_επίσης"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Δείτε επίσης</span> </div> </a> <ul id="toc-Δείτε_επίσης-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Παραπομπές" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Παραπομπές"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Παραπομπές</span> </div> </a> <ul id="toc-Παραπομπές-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div 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">Εγγεγραμμένος και Παρεγγεγραμμένοι κύκλοι 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 29 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-29" 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">29 γλώσσες</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D8%A6%D8%B1_%D8%AF%D8%A7%D8%AE%D9%84%D9%8A%D8%A9_%D9%88%D8%AE%D8%A7%D8%B1%D8%AC%D9%8A%D8%A9_%D9%84%D9%85%D8%AB%D9%84%D8%AB" title="دوائر داخلية وخارجية لمثلث – Αραβικά" lang="ar" hreflang="ar" data-title="دوائر داخلية وخارجية لمثلث" data-language-autonym="العربية" data-language-local-name="Αραβικά" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%B8_%D0%BE%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D0%B8_%D0%B2_%D1%82%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA" title="Вписани окръжности в триъгълник – Βουλγαρικά" lang="bg" hreflang="bg" data-title="Вписани окръжности в триъгълник" 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/Circumfer%C3%A8ncia_inscrita" title="Circumferència inscrita – Καταλανικά" lang="ca" hreflang="ca" data-title="Circumferència inscrita" 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%92%D0%B8%C3%A7%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%C4%95%D1%85%C4%95%D0%BD_%D1%88%D0%B0%D0%BB_%D1%82%D0%B0%D1%82%D0%B0_%C3%A7%D1%83%D0%BC_%C3%A7%D0%B0%D0%B2%D1%80%D0%B0%D0%BA%C4%83%D1%88%C4%95%D1%81%D0%B5%D0%BC" title="Виçкĕтеслĕхĕн шал тата çум çавракăшĕсем – Τσουβασικά" lang="cv" hreflang="cv" data-title="Виçкĕтеслĕхĕн шал тата çум çавракăшĕсем" data-language-autonym="Чӑвашла" data-language-local-name="Τσουβασικά" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mewngylch_ac_allgylch" title="Mewngylch ac allgylch – Ουαλικά" lang="cy" hreflang="cy" data-title="Mewngylch ac allgylch" data-language-autonym="Cymraeg" data-language-local-name="Ουαλικά" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/R%C3%B8ringscirkler" title="Røringscirkler – Δανικά" lang="da" hreflang="da" data-title="Røringscirkler" data-language-autonym="Dansk" data-language-local-name="Δανικά" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Incircle_and_excircles" title="Incircle and excircles – Αγγλικά" lang="en" hreflang="en" data-title="Incircle and excircles" data-language-autonym="English" data-language-local-name="Αγγλικά" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Enskribita_cirklo_kaj_alskribitaj_cirkloj_de_triangulo" title="Enskribita cirklo kaj alskribitaj cirkloj de triangulo – Εσπεράντο" lang="eo" hreflang="eo" data-title="Enskribita cirklo kaj alskribitaj cirkloj de triangulo" data-language-autonym="Esperanto" data-language-local-name="Εσπεράντο" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Circunferencia_inscrita_y_exinscrita_en_un_tri%C3%A1ngulo" title="Circunferencia inscrita y exinscrita en un triángulo – Ισπανικά" lang="es" hreflang="es" data-title="Circunferencia inscrita y exinscrita en un triángulo" 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/Triangeluaren_zirkunferentzia_inskribatua_eta_zirkunferentzia_kanpoinskribatuak" title="Triangeluaren zirkunferentzia inskribatua eta zirkunferentzia kanpoinskribatuak – Βασκικά" lang="eu" hreflang="eu" data-title="Triangeluaren zirkunferentzia inskribatua eta zirkunferentzia kanpoinskribatuak" data-language-autonym="Euskara" data-language-local-name="Βασκικά" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D9%85%D8%AD%D8%A7%D8%B7%DB%8C_%D9%85%D8%AB%D9%84%D8%AB" title="دایره‌های محاطی مثلث – Περσικά" lang="fa" hreflang="fa" data-title="دایره‌های محاطی مثلث" data-language-autonym="فارسی" data-language-local-name="Περσικά" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kolmion_sis%C3%A4%C3%A4n_ja_viereen_piirretyt_ympyr%C3%A4t" title="Kolmion sisään ja viereen piirretyt ympyrät – Φινλανδικά" lang="fi" hreflang="fi" data-title="Kolmion sisään ja viereen piirretyt ympyrät" data-language-autonym="Suomi" data-language-local-name="Φινλανδικά" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cercles_inscrit_et_exinscrits_d%27un_triangle" title="Cercles inscrit et exinscrits d'un triangle – Γαλλικά" lang="fr" hreflang="fr" data-title="Cercles inscrit et exinscrits d'un triangle" data-language-autonym="Français" data-language-local-name="Γαλλικά" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Circunferencia_inscrita_e_exinscrita" title="Circunferencia inscrita e exinscrita – Γαλικιανά" lang="gl" hreflang="gl" data-title="Circunferencia inscrita e exinscrita" data-language-autonym="Galego" data-language-local-name="Γαλικιανά" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C_%E0%A4%95%E0%A5%87_%E0%A4%85%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4_%E0%A4%94%E0%A4%B0_%E0%A4%AC%E0%A4%B9%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="त्रिभुज के अन्तर्वृत्त और बहिर्वृत्त – Χίντι" lang="hi" hreflang="hi" data-title="त्रिभुज के अन्तर्वृत्त और बहिर्वृत्त" data-language-autonym="हिन्दी" data-language-local-name="Χίντι" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/A_h%C3%A1romsz%C3%B6g_be%C3%ADrt_k%C3%B6re_%C3%A9s_hozz%C3%A1%C3%ADrt_k%C3%B6rei" title="A háromszög beírt köre és hozzáírt körei – Ουγγρικά" lang="hu" hreflang="hu" data-title="A háromszög beírt köre és hozzáírt körei" data-language-autonym="Magyar" data-language-local-name="Ουγγρικά" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkaran_dalam_dan_lingkaran_singgung_luar_segitiga" title="Lingkaran dalam dan lingkaran singgung luar segitiga – Ινδονησιακά" lang="id" hreflang="id" data-title="Lingkaran dalam dan lingkaran singgung luar segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="Ινδονησιακά" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E3%81%AE%E5%86%85%E6%8E%A5%E5%86%86%E3%81%A8%E5%82%8D%E6%8E%A5%E5%86%86" title="三角形の内接円と傍接円 – Ιαπωνικά" lang="ja" hreflang="ja" data-title="三角形の内接円と傍接円" 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/%D2%AE%D1%88%D0%B1%D2%B1%D1%80%D1%8B%D1%88%D2%9B%D0%B0_%D1%96%D1%88%D1%82%D0%B5%D0%B9_%D0%B6%D3%99%D0%BD%D0%B5_%D1%81%D1%8B%D1%80%D1%82%D1%82%D0%B0%D0%B9_%D1%81%D1%8B%D0%B7%D1%8B%D0%BB%D2%93%D0%B0%D0%BD_%D1%88%D0%B5%D2%A3%D0%B1%D0%B5%D1%80%D0%BB%D0%B5%D1%80" title="Үшбұрышқа іштей және сырттай сызылған шеңберлер – Καζακικά" lang="kk" hreflang="kk" data-title="Үшбұрышқа іштей және сырттай сызылған шеңберлер" data-language-autonym="Қазақша" data-language-local-name="Καζακικά" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Innvendig_og_utvendig_tangeringssirkel" title="Innvendig og utvendig tangeringssirkel – Νορβηγικά Νινόρσκ" lang="nn" hreflang="nn" data-title="Innvendig og utvendig tangeringssirkel" data-language-autonym="Norsk nynorsk" data-language-local-name="Νορβηγικά Νινόρσκ" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%ADrculos_inscrito_e_exinscrito_em_um_tri%C3%A2ngulo" title="Círculos inscrito e exinscrito em um triângulo – Πορτογαλικά" lang="pt" hreflang="pt" data-title="Círculos inscrito e exinscrito em um triângulo" data-language-autonym="Português" data-language-local-name="Πορτογαλικά" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cerc_%C3%AEnscris_%C8%99i_cerc_ex%C3%AEnscris_unui_triunghi" title="Cerc înscris și cerc exînscris unui triunghi – Ρουμανικά" lang="ro" hreflang="ro" data-title="Cerc înscris și cerc exînscris unui triunghi" data-language-autonym="Română" data-language-local-name="Ρουμανικά" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B8_%D0%B2%D0%BD%D0%B5%D0%B2%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%B2_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA_%D0%BE%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Вписанная и вневписанные в треугольник окружности – Ρωσικά" lang="ru" hreflang="ru" data-title="Вписанная и вневписанные в треугольник окружности" data-language-autonym="Русский" data-language-local-name="Ρωσικά" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/V%C4%8Drtana_in_pri%C4%8Drtana_kro%C5%BEnica_trikotnika" title="Včrtana in pričrtana krožnica trikotnika – Σλοβενικά" lang="sl" hreflang="sl" data-title="Včrtana in pričrtana krožnica trikotnika" data-language-autonym="Slovenščina" data-language-local-name="Σλοβενικά" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Opisana,_upisana_i_spolja_pripisana_kru%C5%BEnica" title="Opisana, upisana i spolja pripisana kružnica – Σερβικά" lang="sr" hreflang="sr" data-title="Opisana, upisana i spolja pripisana kružnica" data-language-autonym="Српски / srpski" data-language-local-name="Σερβικά" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%89%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%81%E0%AE%AE%E0%AF%8D_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%81%E0%AE%AE%E0%AF%8D" title="முக்கோணத்தின் உள்வட்டமும் வெளிவட்டங்களும் – Ταμιλικά" lang="ta" hreflang="ta" data-title="முக்கோணத்தின் உள்வட்டமும் வெளிவட்டங்களும்" data-language-autonym="தமிழ்" data-language-local-name="Ταμιλικά" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1%E0%B9%81%E0%B8%99%E0%B8%9A%E0%B9%83%E0%B8%99%E0%B9%81%E0%B8%A5%E0%B8%B0%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1%E0%B9%81%E0%B8%99%E0%B8%9A%E0%B8%99%E0%B8%AD%E0%B8%81%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1" title="วงกลมแนบในและวงกลมแนบนอกของรูปสามเหลี่ยม – Ταϊλανδικά" lang="th" hreflang="th" data-title="วงกลมแนบในและวงกลมแนบนอกของรูปสามเหลี่ยม" data-language-autonym="ไทย" data-language-local-name="Ταϊλανδικά" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%B5_%D1%96_%D0%B7%D0%BE%D0%B2%D0%BD%D1%96%D0%B2%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%B5_%D0%B2_%D1%82%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA_%D0%BA%D0%BE%D0%BB%D0%B0" title="Вписане і зовнівписане в трикутник кола – Ουκρανικά" lang="uk" hreflang="uk" data-title="Вписане і зовнівписане в трикутник кола" data-language-autonym="Українська" data-language-local-name="Ουκρανικά" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%C3%B2n_n%E1%BB%99i_ti%E1%BA%BFp_v%C3%A0_b%C3%A0ng_ti%E1%BA%BFp" title="Đường tròn nội tiếp và bàng tiếp – Βιετναμικά" lang="vi" hreflang="vi" data-title="Đường tròn nội tiếp và bàng tiếp" data-language-autonym="Tiếng Việt" data-language-local-name="Βιετναμικά" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q10615255#sitelinks-wikipedia" title="Επεξεργασία διαγλωσσικών συνδέσεων" class="wbc-editpage">Επεξεργασία συνδέσμων</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ονοματοχώροι"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><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%CE%B1%CE%B9_%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" title="Προβολή της σελίδας περιεχομένου [c]" accesskey="c"><span>Λήμμα</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%CE%A3%CF%85%CE%B6%CE%AE%CF%84%CE%B7%CF%83%CE%B7:%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%CE%B1%CE%B9_%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&action=edit&redlink=1" rel="discussion" class="new" title="Συζήτηση για τη σελίδα περιεχομένου (δεν έχει γραφτεί ακόμα) [t]" accesskey="t"><span>Συζήτηση</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Αλλαγή παραλλαγής γλώσσας" > 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href="/w/index.php?title=%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%A0%CE%B1%CF%81%CE%B1%CF%80%CE%BF%CE%BC%CF%80%CE%AE%CE%91%CF%85%CF%84%CE%AE%CE%A4%CE%B7%CE%A3%CE%B5%CE%BB%CE%AF%CE%B4%CE%B1&page=%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%CE%B1%CE%B9_%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&id=10794762&wpFormIdentifier=titleform" title="Πληροφορίες για το πώς να δημιουργήσετε παραπομπή αυτής της σελίδας"><span>Παραπομπή</span></a></li><li id="t-urlshortener" class="mw-list-item"><a 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href="/w/index.php?title=%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&redirect=no" class="mw-redirect" title="Παρεγγεγραμμένοι κύκλοι τριγώνου">Παρεγγεγραμμένοι κύκλοι τριγώνου</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="el" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Excircles_and_incircle_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Excircles_and_incircle_el.svg/220px-Excircles_and_incircle_el.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Excircles_and_incircle_el.svg/330px-Excircles_and_incircle_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Excircles_and_incircle_el.svg/440px-Excircles_and_incircle_el.svg.png 2x" data-file-width="443" data-file-height="395" /></a><figcaption>Ο εγγεγραμμένος κύκλος και οι παρεγγεγραμμένοι κύκλοι 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>.</figcaption></figure> <p>Στη <a href="/wiki/%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Γεωμετρία">γεωμετρία</a>, σε ένα <a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο">τρίγωνο</a> ο <b>εγγεγραμμένος κύκλος</b> <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> είναι ο <a href="/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Κύκλος">κύκλος</a> που εφάπτεται εσωτερικά στις τρεις πλευρές του. Το κέντρο του είναι 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 και ονομάζεται <b>έγκεντρο του τριγώνου</b>.<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 class="reference" style="white-space:nowrap;">:80-89</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 class="reference" style="white-space:nowrap;">:143-145</sup><sup id="cite_ref-P74_3-0" class="reference"><a href="#cite_note-P74-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:35-36</sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:12-13</sup> </p><p>Κάθε τρίγωνο έχει επίσης τρεις <b>παρεγγεγραμμένους κύκλους</b> <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 {J_{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">J</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 {J_{A}} ,\rho _{\mathrm {A} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad659770c156d1816fed0f9e49ecd3311af925a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle (\mathrm {J_{A}} ,\rho _{\mathrm {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 (\mathrm {J_{B}} ,\rho _{\mathrm {B} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {J_{B}} ,\rho _{\mathrm {B} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2bd3d6ade6a8a07c279d09b0c71d69f397dcc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.032ex; height:2.843ex;" alt="{\displaystyle (\mathrm {J_{B}} ,\rho _{\mathrm {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 {J_{\Gamma }} ,\rho _{\mathrm {\Gamma } })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {J_{\Gamma }} ,\rho _{\mathrm {\Gamma } })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37704f3b9703ac34cf7393fe22c10e10853db39e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="{\displaystyle (\mathrm {J_{\Gamma }} ,\rho _{\mathrm {\Gamma } })}"></span> που εφάπτονται στις τρεις πλευρές του τριγώνου εξωτερικά αυτού. Το κέντρο 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 (\mathrm {J_{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">J</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 {J_{A}} ,\rho _{\mathrm {A} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad659770c156d1816fed0f9e49ecd3311af925a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle (\mathrm {J_{A}} ,\rho _{\mathrm {A} })}"></span> είναι το σημείο τομής της διχοτόμου 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 {\hat {\rm {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 {\rm {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7842b91cfc233938833a9f83281638fbb05fb257" 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 {\rm {A}}}}"></span> και των <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">εξωτερικών διχοτόμων</a> 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 {\hat {\rm {B}}}}"> <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> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\rm {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccdb4379a54b53e44aa58356f946e1d5175a1b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\hat {\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 {\hat {\rm {\Gamma }}}}"> <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> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\rm {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e3c2f6e3235ee11ebd66f03273c3ae0cc2f0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\hat {\rm {\Gamma }}}}"></span>, και ονομάζεται <b>παράκεντρο του τριγώνου</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Εγγεγραμμένος_κύκλος"><span id=".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"></span>Εγγεγραμμένος κύκλος</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=1" title="Επεξεργασία ενότητας: Εγγεγραμμένος κύκλος" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&action=edit&section=1" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Εγγεγραμμένος κύκλος"><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>Οι εσωτερικές διχοτόμοι <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\Delta _{A},B\Delta _{B},\Gamma \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">A</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Delta _{A},B\Delta _{B},\Gamma \Delta _{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af8116d06ea30580b3ac2244507bb96aef8209a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.837ex; height:2.509ex;" alt="{\displaystyle {\rm {A\Delta _{A},B\Delta _{B},\Gamma \Delta _{\Gamma }}}}"></span> ενός τριγώνου διέρχονται από το ίδιο σημείο, το έγκεντρο, το οποίο είναι το κέντρου του εγγεγραμμένου κύκλου. </p> </div> <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:Incircle_for_acute_triangle_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Incircle_for_acute_triangle_el.svg/200px-Incircle_for_acute_triangle_el.svg.png" decoding="async" width="200" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Incircle_for_acute_triangle_el.svg/300px-Incircle_for_acute_triangle_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Incircle_for_acute_triangle_el.svg/400px-Incircle_for_acute_triangle_el.svg.png 2x" data-file-width="208" data-file-height="143" /></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:Incircle_of_orthogonal_triangle_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Incircle_of_orthogonal_triangle_el.svg/200px-Incircle_of_orthogonal_triangle_el.svg.png" decoding="async" width="200" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Incircle_of_orthogonal_triangle_el.svg/300px-Incircle_of_orthogonal_triangle_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Incircle_of_orthogonal_triangle_el.svg/400px-Incircle_of_orthogonal_triangle_el.svg.png 2x" data-file-width="208" data-file-height="143" /></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:Incircle_for_obtuse_triangle_el.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Incircle_for_obtuse_triangle_el.svg/200px-Incircle_for_obtuse_triangle_el.svg.png" decoding="async" width="200" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Incircle_for_obtuse_triangle_el.svg/300px-Incircle_for_obtuse_triangle_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Incircle_for_obtuse_triangle_el.svg/400px-Incircle_for_obtuse_triangle_el.svg.png 2x" data-file-width="251" data-file-height="98" /></a></span></div></div><div style="clear:left"></div><div class="thumbcaption" style="clear:left;text-align:center;background-color:transparent">Το έγκεντρο και ο εγγεγραμμένος κύκλος σε ένα οξυγώνιο, ένα ορθογώνιο και ένα αμβλυγώνιο τρίγωνο.</div></div></div> <div class="mw-heading mw-heading3"><h3 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</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=2" 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%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%CE%B1%CE%B9_%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&action=edit&section=2" 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: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 ιδιότητες διχοτόμων)</strong>   </td></tr> <tr> <td> <p>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 {B\Delta _{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">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Delta _{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e43afa34a2de5881ef41b9d2ca2681e5eea3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.977ex; height:2.509ex;" alt="{\displaystyle {\rm {B\Delta _{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 \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">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b396eb202396d8cf58dbc7602fd4b9d44447f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.648ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}"></span> οι διχοτόμοι των γωνιών <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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></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 {I'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">I</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb85b001838601e633d8cf06a4b70084600171e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.524ex; height:2.509ex;" alt="{\displaystyle {\rm {I'}}}"></span> το σημείο τομής τους. Από την κύρια ιδιότητα των διχοτόμων, κάθε σημείο της <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\Delta _{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">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Delta _{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e43afa34a2de5881ef41b9d2ca2681e5eea3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.977ex; height:2.509ex;" alt="{\displaystyle {\rm {B\Delta _{B}}}}"></span> ισαπέχει από τις πλευρές <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 {BA}}}"> <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">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BA}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971571fb20de3e12de46f70ad075d6f7930d5489" 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 {BA}}}"></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 {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>. Αντίστοιχα, για 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 {\rm {\Gamma \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">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b396eb202396d8cf58dbc7602fd4b9d44447f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.648ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}"></span>. </p><p>Επομένως, 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 {I'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">I</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb85b001838601e633d8cf06a4b70084600171e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.524ex; height:2.509ex;" alt="{\displaystyle {\rm {I'}}}"></span> ισαπέχει από τις τρεις πλευρές του τριγώνου και άρα ανήκει και στην διχοτόμου της <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 {\hat {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba475e78125ade3491258fde2f9efc4dc9f8afe0" 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 {\rm {\hat {A}}}}"></span>. Καταλήγουμε 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 {\rm {I'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">I</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb85b001838601e633d8cf06a4b70084600171e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.524ex; height:2.509ex;" alt="{\displaystyle {\rm {I'}}}"></span> είναι το σημείο τομής των διχοτόμων του τριγώνου. </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 Τσέβα)</strong>   </td></tr> <tr> <td> <p>Από το <a href="/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%B5%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CE%AE%CF%82_%CE%B4%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%85" class="mw-redirect" title="Θεώρημα εσωτερικής διχοτόμου">θεώρημα εσωτερικής διχοτόμου</a> έχουμε για τις διχοτόμους <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\Delta _{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">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Delta _{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3616ceb2ead0744bf599502a146f9e81eb87e112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.144ex; height:2.509ex;" alt="{\displaystyle {\rm {A\Delta _{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 {B\Delta _{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">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Delta _{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e43afa34a2de5881ef41b9d2ca2681e5eea3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.977ex; height:2.509ex;" alt="{\displaystyle {\rm {B\Delta _{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 \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">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b396eb202396d8cf58dbc7602fd4b9d44447f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.648ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma \Delta _{\Gamma }}}}"></span> 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 {{\frac {AB}{A\Gamma }}={\frac {B\Delta _{A}}{\Gamma \Delta _{A}}},\quad {\rm {{\frac {BA}{B\Gamma }}={\frac {A\Delta _{B}}{\Gamma \Delta _{B}}}\quad }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mrow> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {{\frac {AB}{A\Gamma }}={\frac {B\Delta _{A}}{\Gamma \Delta _{A}}},\quad {\rm {{\frac {BA}{B\Gamma }}={\frac {A\Delta _{B}}{\Gamma \Delta _{B}}}\quad }}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e069a733ea36dc4251603859fc9b0176071605af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.119ex; height:5.843ex;" alt="{\displaystyle {\rm {{\frac {AB}{A\Gamma }}={\frac {B\Delta _{A}}{\Gamma \Delta _{A}}},\quad {\rm {{\frac {BA}{B\Gamma }}={\frac {A\Delta _{B}}{\Gamma \Delta _{B}}}\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 {\rm {{\frac {\Gamma A}{\Gamma B}}={\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\rm {{\frac {\Gamma A}{\Gamma B}}={\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ce5aaf33ce28fd47cf358af09532dc0033a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.875ex; height:5.843ex;" alt="{\displaystyle \quad {\rm {{\frac {\Gamma A}{\Gamma B}}={\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}.}}}"></span></dd></dl> <p>Επομένως, 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 {{\frac {B\Delta _{A}}{\Gamma \Delta _{A}}}\cdot {\frac {\Gamma \Delta _{B}}{A\Delta _{B}}}\cdot {\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}={\frac {AB}{A\Gamma }}\cdot {\frac {B\Gamma }{BA}}\cdot {\frac {\Gamma A}{\Gamma B}}=1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">Δ</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> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">Δ</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> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {{\frac {B\Delta _{A}}{\Gamma \Delta _{A}}}\cdot {\frac {\Gamma \Delta _{B}}{A\Delta _{B}}}\cdot {\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}={\frac {AB}{A\Gamma }}\cdot {\frac {B\Gamma }{BA}}\cdot {\frac {\Gamma A}{\Gamma B}}=1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268e2c0e933956373ffea646c40177e1784be3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.125ex; height:5.843ex;" alt="{\displaystyle {\rm {{\frac {B\Delta _{A}}{\Gamma \Delta _{A}}}\cdot {\frac {\Gamma \Delta _{B}}{A\Delta _{B}}}\cdot {\frac {A\Delta _{\Gamma }}{B\Delta _{\Gamma }}}={\frac {AB}{A\Gamma }}\cdot {\frac {B\Gamma }{BA}}\cdot {\frac {\Gamma A}{\Gamma B}}=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> καταλήγουμε ότι οι τρεις διχοτόμοι διέρχονται από το ίδιο σημείο. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ιδιότητες"><span id=".CE.99.CE.B4.CE.B9.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82"></span>Ιδιότητες</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=3" 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%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%CE%B1%CE%B9_%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&action=edit&section=3" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ιδιότητες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Το έγκεντρο <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 {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7143a9ca5e817630342a8959f5673b0d2895964d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.176ex;" alt="{\displaystyle {\rm {I}}}"></span> είναι σημείο εσωτερικό του τριγώνου.</li> <li>Η γωνία των διχοτόμων 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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></span> είναι ίση με <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 90^{o}+{\tfrac {\hat {\rm {A}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{o}+{\tfrac {\hat {\rm {A}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362c421ea720ebd993f53de999afdc27b0ada04c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.264ex; height:4.176ex;" alt="{\displaystyle 90^{o}+{\tfrac {\hat {\rm {A}}}{2}}}"></span>.<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: 85">: 85 </span></sup></li></ul> <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</strong>   </td></tr> <tr> <td> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Angle_between_bisectors_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Angle_between_bisectors_el.svg/220px-Angle_between_bisectors_el.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Angle_between_bisectors_el.svg/330px-Angle_between_bisectors_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Angle_between_bisectors_el.svg/440px-Angle_between_bisectors_el.svg.png 2x" data-file-width="208" data-file-height="143" /></a><figcaption>Σχήμα απόδειξης</figcaption></figure> <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 {\rm {BI}}}"> <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">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50924ca10ee8aedcd9b84ac33c4c2ef2035f5e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.485ex; height:2.176ex;" alt="{\displaystyle {\rm {BI}}}"></span> η διχοτόμος της <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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {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 I}}}"> <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">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cca0aa11766f2c273d25d8d4e234c881bd84a37f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.292ex; height:2.176ex;" alt="{\displaystyle {\rm {\Gamma I}}}"></span> η διχοτόμος της <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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></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 {\rm {\angle IB\Gamma ={\tfrac {\hat {B}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle IB\Gamma ={\tfrac {\hat {B}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/394e2b58d65db414641d22a8de3e3064e9ac1d10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.714ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle IB\Gamma ={\tfrac {\hat {B}}{2}}}}}"></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 {\angle I\Gamma B={\tfrac {\hat {\Gamma }}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle I\Gamma B={\tfrac {\hat {\Gamma }}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204843e2a649ace2fd719cc45a9d917c7fc3957d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.578ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle I\Gamma B={\tfrac {\hat {\Gamma }}{2}}}}}"></span>. Επομένως, αφού οι γωνίες 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 {BI\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">I</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8f6547b08633d45aee4f44cfc6df74bd2a1eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.938ex; height:2.176ex;" alt="{\displaystyle {\rm {BI\Gamma }}}"></span> αθροίζουν στις <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 180^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70447d0ef4aba9cab93645545f8f40665c914d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.517ex; height:2.343ex;" alt="{\displaystyle 180^{o}}"></span> </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 {\angle AIB=180^{o}-{\frac {{\hat {B}}+{\hat {\Gamma }}}{2}}=180^{o}-{\frac {180^{o}-{\hat {A}}}{2}}=90^{0}+{\tfrac {\hat {A}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">B</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle AIB=180^{o}-{\frac {{\hat {B}}+{\hat {\Gamma }}}{2}}=180^{o}-{\frac {180^{o}-{\hat {A}}}{2}}=90^{0}+{\tfrac {\hat {A}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b51e85a3140f07458c896230a49c8d6815bebef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.99ex; height:6.009ex;" alt="{\displaystyle {\rm {\angle AIB=180^{o}-{\frac {{\hat {B}}+{\hat {\Gamma }}}{2}}=180^{o}-{\frac {180^{o}-{\hat {A}}}{2}}=90^{0}+{\tfrac {\hat {A}}{2}}}}}"></span>.</dd></dl> </td></tr></tbody></table> <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 {\rm {I_{A},I_{B},I_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A},I_{B},I_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25ee7e1fac9562be33220bccabd24119cc612d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.707ex; height:2.509ex;" alt="{\displaystyle {\rm {I_{A},I_{B},I_{\Gamma }}}}"></span> οι προβολές 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 {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7143a9ca5e817630342a8959f5673b0d2895964d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.176ex;" alt="{\displaystyle {\rm {I}}}"></span> στις πλευρές 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 {\rm {BI_{A}=BI_{\Gamma }=\tau -\beta ,\quad AI_{B}=AI_{\Gamma }=\tau -\alpha ,\quad }}}"> <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">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo>,</mo> <mspace width="1em" /> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI_{A}=BI_{\Gamma }=\tau -\beta ,\quad AI_{B}=AI_{\Gamma }=\tau -\alpha ,\quad }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c7bed7e3aff85ba82038f8e3319f714f4b2c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.14ex; height:2.509ex;" alt="{\displaystyle {\rm {BI_{A}=BI_{\Gamma }=\tau -\beta ,\quad AI_{B}=AI_{\Gamma }=\tau -\alpha ,\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 {\rm {\Gamma I_{A}=\Gamma I_{B}=\tau -\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\rm {\Gamma I_{A}=\Gamma I_{B}=\tau -\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0964ceae8e897a17f2bbacd4714edb6562638130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.27ex; height:2.676ex;" alt="{\displaystyle \quad {\rm {\Gamma I_{A}=\Gamma I_{B}=\tau -\gamma }}}"></span>.</dd></dl></dd></dl> <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</strong>   </td></tr> <tr> <td> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Incircle_projections_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Incircle_projections_el.svg/220px-Incircle_projections_el.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Incircle_projections_el.svg/330px-Incircle_projections_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Incircle_projections_el.svg/440px-Incircle_projections_el.svg.png 2x" data-file-width="208" data-file-height="145" /></a><figcaption>Σχήμα απόδειξης</figcaption></figure> <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 {\rm {AII_{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">I</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AII_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aae951fac73a5d8ef3e7d0a1b4a1f8630e54046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.818ex; height:2.509ex;" alt="{\displaystyle {\rm {AII_{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 {AII_{\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">I</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AII_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1267ee44008a6a4c2173021554e1a0739b019173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.682ex; height:2.509ex;" alt="{\displaystyle {\rm {AII_{\Gamma }}}}"></span> είναι ίσα, 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 {AI}}}"> <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">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7ffeddaf0345dd0f6cd62a427cc2e54b35610d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle {\rm {AI}}}"></span> κοινή, <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 {\angle IAI_{B}=\angle IAI_{\Gamma }={\tfrac {\hat {A}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle IAI_{B}=\angle IAI_{\Gamma }={\tfrac {\hat {A}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c090113f1c8183f60436210ef67d87ee10e765b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.122ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle IAI_{B}=\angle IAI_{\Gamma }={\tfrac {\hat {A}}{2}}}}}"></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 {{\hat {I_{B}}}={\hat {I_{\Gamma }}}=90^{o}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {{\hat {I_{B}}}={\hat {I_{\Gamma }}}=90^{o}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55ec644593643cfb0b31f2b7cf78047f9975c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.911ex; height:3.176ex;" alt="{\displaystyle {\rm {{\hat {I_{B}}}={\hat {I_{\Gamma }}}=90^{o}}}}"></span> (<a href="/wiki/%CE%8A%CF%83%CE%B1_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%B1#Κριτήριο_γωνίας-πλευράς-γωνίας" title="Ίσα τρίγωνα">κριτήριο Γ-Π-Γ</a>). Επομένως, <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 {AI_{B}=AI_{\Gamma }=x}}}"> <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">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">x</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{B}=AI_{\Gamma }=x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd92acef0e96a168798aa7cf4f2ccf4b67f1ad7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.245ex; height:2.509ex;" alt="{\displaystyle {\rm {AI_{B}=AI_{\Gamma }=x}}}"></span>. Αντίστοιχα, <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 {BI_{A}=B_{\Gamma }=y}}}"> <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">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">y</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI_{A}=B_{\Gamma }=y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193bbdd0d6feb8a9fe47e2e06568fde42dae099f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.279ex; height:2.509ex;" alt="{\displaystyle {\rm {BI_{A}=B_{\Gamma }=y}}}"></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 I_{A}=\Gamma I_{B}=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> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">z</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma I_{A}=\Gamma I_{B}=z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389d62b6be69f9e50531e55e02f8fa104278f1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.675ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma I_{A}=\Gamma I_{B}=z}}}"></span>. </p><p>Επίσης, 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 {\begin{aligned}y+z&=\alpha \\x+z&=\beta \\y+x&=\gamma \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>+</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y+z&=\alpha \\x+z&=\beta \\y+x&=\gamma \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6efb5da2335121fa3107cf63d1aab0b8b6fc5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:10.663ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}y+z&=\alpha \\x+z&=\beta \\y+x&=\gamma \end{aligned}}}"></span></dd></dl> <p>Λύνοντας το σύστημα των εξισώσεων βρίσκουμε 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 x=\tau -\alpha ,\quad y=\tau -\beta \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\tau -\alpha ,\quad y=\tau -\beta \quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08438319c3d31d08c9536e062910ece9565d5987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.266ex; height:2.509ex;" alt="{\displaystyle x=\tau -\alpha ,\quad y=\tau -\beta \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 z=\tau -\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\tau -\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf301e52520e85f59c8a3760fa87732302cc4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.491ex; height:2.509ex;" alt="{\displaystyle z=\tau -\gamma }"></span>,</dd></dl> <p>όπου <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 \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4834dcbe34b65edd6abdf2b075171b33a2c68814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.21ex; height:3.509ex;" alt="{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"></span>. </p> </td></tr></tbody></table> <ul><li>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 {I_{A}I_{B}I_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A}I_{B}I_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ee18e3b266e928532795ad9c134beeede2aa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.639ex; height:2.509ex;" alt="{\displaystyle {\rm {I_{A}I_{B}I_{\Gamma }}}}"></span> ονομάζεται το <a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF_Gergonne" title="Τρίγωνο Gergonne">τρίγωνο Gergonne</a>.</li> <li>(<b><a href="/wiki/%CE%A3%CE%B7%CE%BC%CE%B5%CE%AF%CE%BF_Gergonne" title="Σημείο Gergonne">Σημείο Gergonne</a></b>) Τα ευθύγραμμα τμήματα <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 {AI_{A},BI_{B},\Gamma I_{\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> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{A},BI_{B},\Gamma I_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d943751077ef54e12baae86a09853e4fa82cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.548ex; height:2.509ex;" alt="{\displaystyle {\rm {AI_{A},BI_{B},\Gamma I_{\Gamma }}}}"></span> διέρχονται από το ίδιο σημείο.<sup id="cite_ref-P74_3-1" class="reference"><a href="#cite_note-P74-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 36">: 36 </span></sup></li> <li>Οι ευθείες <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 {IA,IB,I\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">I</mi> <mi mathvariant="normal">A</mi> <mo>,</mo> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">B</mi> <mo>,</mo> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {IA,IB,I\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a775a81d36ada295b41fc21723a48c5482c68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.428ex; height:2.509ex;" alt="{\displaystyle {\rm {IA,IB,I\Gamma }}}"></span> είναι <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">μεσοκάθετοι</a> των πλευρών 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 {I_{A}I_{B}I_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A}I_{B}I_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ee18e3b266e928532795ad9c134beeede2aa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.639ex; height:2.509ex;" alt="{\displaystyle {\rm {I_{A}I_{B}I_{\Gamma }}}}"></span>.</li> <li>Το <a href="/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD" title="Εμβαδόν">εμβαδόν</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> δίνεται από τον τύπο <sup id="cite_ref-Tog_5-0" class="reference"><a href="#cite_note-Tog-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:126</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 {E} =\tau \cdot \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} =\tau \cdot \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3a60b7d0753c18867fced29c272d6cb8c982e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.764ex; height:2.676ex;" alt="{\displaystyle \mathrm {E} =\tau \cdot \rho }"></span>,</dd></dl></dd> <dd>όπου <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 \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4834dcbe34b65edd6abdf2b075171b33a2c68814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.21ex; height:3.509ex;" alt="{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"></span> είναι η <a href="/wiki/%CE%97%CE%BC%CE%B9%CF%80%CE%B5%CF%81%CE%AF%CE%BC%CE%B5%CF%84%CF%81%CE%BF%CF%82" title="Ημιπερίμετρος">ημιπερίμετρος</a> του τριγώνου.</dd></dl> <ul><li>Από τον <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">τύπο του Ήρωνα</a>, προκύπτει ότι η ακτίνα του εγγεγραμμένου κύκλου<sup id="cite_ref-K75_6-0" class="reference"><a href="#cite_note-K75-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:139</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 \rho ={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mi>τ</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0654dc414df0e9b61d6b130f0a0ae385fbd6323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.455ex; height:7.676ex;" alt="{\displaystyle \rho ={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau }}}}"></span>.</dd></dl></dd></dl> <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</strong>   </td></tr> <tr> <td> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Incircle_area_proof_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/el/thumb/7/77/Incircle_area_proof_el.svg/220px-Incircle_area_proof_el.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/el/thumb/7/77/Incircle_area_proof_el.svg/330px-Incircle_area_proof_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/el/thumb/7/77/Incircle_area_proof_el.svg/440px-Incircle_area_proof_el.svg.png 2x" data-file-width="195" data-file-height="136" /></a><figcaption>Σχήμα απόδειξης</figcaption></figure> <p>Το εμβαδόν 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> δίνεται από τύπο </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 {E_{AB\Gamma }=E_{ABI}+E_{B\Gamma I}+E_{\Gamma AI}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{AB\Gamma }=E_{ABI}+E_{B\Gamma I}+E_{\Gamma AI}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ceec4209f75f1d255d25f752668163df21a56fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.091ex; height:2.509ex;" alt="{\displaystyle {\rm {E_{AB\Gamma }=E_{ABI}+E_{B\Gamma I}+E_{\Gamma AI}}}}"></span>.</dd></dl> <p>Από τον τύπο για το εμβαδόν του τριγώνου 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 {E_{ABI}}}={\tfrac {1}{2}}\cdot \gamma \cdot \rho ,\quad {\rm {E_{B\Gamma I}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho ,\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>γ</mi> <mo>⋅</mo> <mi>ρ</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>α</mi> <mo>⋅</mo> <mi>ρ</mi> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{ABI}}}={\tfrac {1}{2}}\cdot \gamma \cdot \rho ,\quad {\rm {E_{B\Gamma I}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho ,\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bdf28e5df87f7babecfbd0a846cdd62812df2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.114ex; height:3.509ex;" alt="{\displaystyle {\rm {E_{ABI}}}={\tfrac {1}{2}}\cdot \gamma \cdot \rho ,\quad {\rm {E_{B\Gamma I}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho ,\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 {\rm {E_{\Gamma AI}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">I</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>α</mi> <mo>⋅</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\rm {E_{\Gamma AI}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e260fd0c854fb8919265ef57ec50bb76d966dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.796ex; height:3.509ex;" alt="{\displaystyle \quad {\rm {E_{\Gamma AI}}}={\tfrac {1}{2}}\cdot \alpha \cdot \rho }"></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 {\rm {E_{AB\Gamma }={\tfrac {1}{2}}\cdot \alpha \cdot \rho +{\tfrac {1}{2}}\cdot \beta \cdot \rho +{\tfrac {1}{2}}\cdot \gamma \cdot \rho ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )\cdot \rho =\tau \cdot \rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>α</mi> <mo>⋅</mo> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>β</mi> <mo>⋅</mo> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>γ</mi> <mo>⋅</mo> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </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> <mi>ρ</mi> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>ρ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{AB\Gamma }={\tfrac {1}{2}}\cdot \alpha \cdot \rho +{\tfrac {1}{2}}\cdot \beta \cdot \rho +{\tfrac {1}{2}}\cdot \gamma \cdot \rho ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )\cdot \rho =\tau \cdot \rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647416757ad91601acd0a703de3d9aabd39f089c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:64.825ex; height:3.509ex;" alt="{\displaystyle {\rm {E_{AB\Gamma }={\tfrac {1}{2}}\cdot \alpha \cdot \rho +{\tfrac {1}{2}}\cdot \beta \cdot \rho +{\tfrac {1}{2}}\cdot \gamma \cdot \rho ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )\cdot \rho =\tau \cdot \rho }}}"></span>.</dd></dl> <p>Από τον τύπο του Ήρωνα, προκύπτει ο πρώτος τύπος για το εμβαδόν. </p> </td></tr></tbody></table> <ul><li>Επίσης, η ακτίνα του εγγεγραμμένου κύκλου του τριγώνου, δίνεται από τις εξής <a href="/wiki/%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Τριγωνομετρία">τριγωνομετρικές σχέσεις</a><sup id="cite_ref-P57_7-0" class="reference"><a href="#cite_note-P57-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 261-262">: 261-262 </span></sup><sup id="cite_ref-Tog_5-1" class="reference"><a href="#cite_note-Tog-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 126">: 126 </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 \rho =\alpha \cdot {\frac {\sin {\frac {\rm {B}}{2}}\cdot \sin {\frac {\Gamma }{2}}}{\cos {\frac {\rm {A}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\rm {\Gamma }}{2}}\cdot \sin {\frac {A}{2}}}{\cos {\frac {\rm {B}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\rm {A}}{2}}\cdot \sin {\frac {B}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\alpha \cdot {\frac {\sin {\frac {\rm {B}}{2}}\cdot \sin {\frac {\Gamma }{2}}}{\cos {\frac {\rm {A}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\rm {\Gamma }}{2}}\cdot \sin {\frac {A}{2}}}{\cos {\frac {\rm {B}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\rm {A}}{2}}\cdot \sin {\frac {B}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49ff5e4c8a102151349e3ffdceaaf5519405836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:58.566ex; height:8.176ex;" alt="{\displaystyle \rho =\alpha \cdot {\frac {\sin {\frac {\rm {B}}{2}}\cdot \sin {\frac {\Gamma }{2}}}{\cos {\frac {\rm {A}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\rm {\Gamma }}{2}}\cdot \sin {\frac {A}{2}}}{\cos {\frac {\rm {B}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\rm {A}}{2}}\cdot \sin {\frac {B}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}"></span>,</dd></dl></dd> <dd>και από <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 \rho =(\tau -\alpha )\cdot \tan {\frac {\rm {A}}{2}}=(\tau -\beta )\cdot \tan {\frac {\rm {B}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\rm {\Gamma }}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =(\tau -\alpha )\cdot \tan {\frac {\rm {A}}{2}}=(\tau -\beta )\cdot \tan {\frac {\rm {B}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\rm {\Gamma }}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6982a1f7d2d22270ebfabcc07a8a3a889f59c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.761ex; height:5.343ex;" alt="{\displaystyle \rho =(\tau -\alpha )\cdot \tan {\frac {\rm {A}}{2}}=(\tau -\beta )\cdot \tan {\frac {\rm {B}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\rm {\Gamma }}{2}}}"></span>.</dd></dl></dd></dl> <ul><li>(<b><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></b>) 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 {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> είναι 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 κύκλος</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 {OI} ^{2}=R^{2}-2R\rho }"> <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">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>R</mi> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {OI} ^{2}=R^{2}-2R\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0023958caf5f5d909ef51b7c8d37d1bff8b6c071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.587ex; height:3.176ex;" alt="{\displaystyle \mathrm {OI} ^{2}=R^{2}-2R\rho }"></span>.</dd></dl></dd></dl> <ul><li>(<b><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</a></b>) 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 {\rm {OM_{A},OM_{B},OM_{\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">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {OM_{A},OM_{B},OM_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c38e3c8dd209f99ac76cbc73a1b5b27c55fc33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.005ex; height:2.509ex;" alt="{\displaystyle {\rm {OM_{A},OM_{B},OM_{\Gamma }}}}"></span> είναι οι προσημασμένες αποστάσεις του περίκεντρου από τις πλευρές 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> 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 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 περιγεγραμμένου κύκλου, 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 {\rm {OM_{A}+OM_{B}+OM_{\Gamma }}}=R+\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {OM_{A}+OM_{B}+OM_{\Gamma }}}=R+\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00597fd8c07c147d7a1043e89a72012294ae1714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.523ex; height:2.676ex;" alt="{\displaystyle {\rm {OM_{A}+OM_{B}+OM_{\Gamma }}}=R+\rho }"></span>.</dd></dl></dd></dl> <ul><li>Οι τριγραμμικές συντεταγμένες του έγκεντρου 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 1:1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f807f34ce5a0799a9c736a216f231aab754a2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.362ex; height:2.176ex;" alt="{\displaystyle 1:1:1}"></span>.</li> <li>Οι βαρυκεντρικές συντεταγμένες του έγκεντρου 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 \alpha :\beta :\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.676ex;" alt="{\displaystyle \alpha :\beta :\gamma }"></span>.</li> <li>Οι καρτεσιανές συντεταγμένες του έγκεντρου 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 \left({\frac {\alpha \cdot x_{\rm {A}}+\beta \cdot x_{\rm {B}}+\gamma \cdot x_{\rm {\Gamma }}}{\alpha +\beta +\gamma }},{\frac {\alpha \cdot y_{\rm {A}}+\beta \cdot y_{\rm {B}}+\gamma \cdot y_{\rm {\Gamma }}}{\alpha +\beta +\gamma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mo>⋅</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mo>⋅</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\alpha \cdot x_{\rm {A}}+\beta \cdot x_{\rm {B}}+\gamma \cdot x_{\rm {\Gamma }}}{\alpha +\beta +\gamma }},{\frac {\alpha \cdot y_{\rm {A}}+\beta \cdot y_{\rm {B}}+\gamma \cdot y_{\rm {\Gamma }}}{\alpha +\beta +\gamma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cd9ec1fc0ce5bfe1d8eb485edcd9ac61bb7c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.375ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\alpha \cdot x_{\rm {A}}+\beta \cdot x_{\rm {B}}+\gamma \cdot x_{\rm {\Gamma }}}{\alpha +\beta +\gamma }},{\frac {\alpha \cdot y_{\rm {A}}+\beta \cdot y_{\rm {B}}+\gamma \cdot y_{\rm {\Gamma }}}{\alpha +\beta +\gamma }}\right)}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Παρεγγεγραμμένοι_κύκλοι"><span id=".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"></span>Παρεγγεγραμμένοι κύκλοι</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=4" title="Επεξεργασία ενότητας: Παρεγγεγραμμένοι κύκλοι" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&action=edit&section=4" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Παρεγγεγραμμένοι κύκλοι"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 {\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> έχει τρεις παρεγγεγραμμένους κύκλους <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 {J_{A}}},\rho _{\rm {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </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 ({\rm {J_{A}}},\rho _{\rm {A}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d76c85ab0c126204cc5840e699bcde22ef25a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle ({\rm {J_{A}}},\rho _{\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 {J_{B}}},\rho _{\rm {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {J_{B}}},\rho _{\rm {B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2505803fe8cc5e3c04a3832ca5fa8c5c4e4af4cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.032ex; height:2.843ex;" alt="{\displaystyle ({\rm {J_{B}}},\rho _{\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 {J_{\Gamma }}},\rho _{\rm {\Gamma }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {J_{\Gamma }}},\rho _{\rm {\Gamma }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d97b99b77ae222eb7cd9dc5a8f0897ff4538e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="{\displaystyle ({\rm {J_{\Gamma }}},\rho _{\rm {\Gamma }})}"></span>. Ο παρεγγεγραμμένος κύκλος <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 {J_{A}}},\rho _{\rm {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </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 ({\rm {J_{A}}},\rho _{\rm {A}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d76c85ab0c126204cc5840e699bcde22ef25a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle ({\rm {J_{A}}},\rho _{\rm {A}})}"></span> έχει κέντρο το σημείο τομής των εξωτερικών διχοτόμων της γωνίας <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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {B}}}}"></span> και της <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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></span> και της εσωτερικής διχοτόμου της <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 {\hat {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba475e78125ade3491258fde2f9efc4dc9f8afe0" 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 {\rm {\hat {A}}}}"></span>. Τα σημεία που εφάπτεται ο κύκλος <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 {J_{A}}},\rho _{\rm {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </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 ({\rm {J_{A}}},\rho _{\rm {A}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d76c85ab0c126204cc5840e699bcde22ef25a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle ({\rm {J_{A}}},\rho _{\rm {A}})}"></span> με τις πλευρές <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 ,AB,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">B</mi> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mo>,</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Gamma ,AB,A\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b84a4e31c795b6117ca7708aeb6911f5f9d91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.751ex; height:2.509ex;" alt="{\displaystyle {\rm {B\Gamma ,AB,A\Gamma }}}"></span> συμβολίζονται με <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 {I_{A}',I_{A}'',I_{A}'''}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>,</mo> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A}',I_{A}'',I_{A}'''}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689c5d66577789cce1a6d4f17d8c3cadcec3bdaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.106ex; height:2.843ex;" alt="{\displaystyle {\rm {I_{A}',I_{A}'',I_{A}'''}}}"></span> αντίστοιχα. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Excircles_triangle_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Excircles_triangle_el.svg/350px-Excircles_triangle_el.svg.png" decoding="async" width="350" height="312" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Excircles_triangle_el.svg/525px-Excircles_triangle_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Excircles_triangle_el.svg/700px-Excircles_triangle_el.svg.png 2x" data-file-width="443" data-file-height="395" /></a><figcaption>Οι παρεγγεγραμμένοι κύκλοι 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>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Απόδειξη"><span id=".CE.91.CF.80.CF.8C.CE.B4.CE.B5.CE.B9.CE.BE.CE.B7"></span>Απόδειξη</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=5" title="Επεξεργασία ενότητας: Απόδειξη" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&action=edit&section=5" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Απόδειξη"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 {J_{\Gamma }'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{\Gamma }'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebab62c98c4530498c3376227a85d4e188d0808a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.455ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{\Gamma }'}}}"></span> το σημείο τομής των εξωτερικών διχοτόμων 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 {\hat {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba475e78125ade3491258fde2f9efc4dc9f8afe0" 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 {\rm {\hat {A}}}}"></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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {B}}}}"></span>. Τότε, από την ιδιότητα της διχοτόμου, όλα τα σημεία ισαπέχουν από τις πλευρές της, άρα <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 {J_{A}I_{A}''=J_{A}I_{A}'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}I_{A}''=J_{A}I_{A}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae588766d84e0087037437fbae5dbeb001ca5899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.027ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{A}I_{A}''=J_{A}I_{A}'}}}"></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 {J_{A}I_{A}'''=J_{A}I_{A}'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}I_{A}'''=J_{A}I_{A}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db852406f0ce9465ae0da0619a813ab3890f01be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.152ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{A}I_{A}'''=J_{A}I_{A}'}}}"></span>. Επομένως, <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 {J_{A}I_{A}''=J_{A}I_{A}'''}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}I_{A}''=J_{A}I_{A}'''}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b1526203803be0f4652c9da42276007f3041ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.152ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{A}I_{A}''=J_{A}I_{A}'''}}}"></span> και έτσι 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 {I_{A}'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67003376281a40bc066ad7f42d1a939aca652359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.304ex; height:2.843ex;" alt="{\displaystyle {\rm {I_{A}'}}}"></span> είναι σημείο της διχοτόμου 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 {\hat {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba475e78125ade3491258fde2f9efc4dc9f8afe0" 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 {\rm {\hat {A}}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ιδιότητες_2"><span id=".CE.99.CE.B4.CE.B9.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82_2"></span>Ιδιότητες</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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%CE%B1%CE%B9_%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&veaction=edit&section=6" 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%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%CE%B1%CE%B9_%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&action=edit&section=6" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ιδιότητες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Τα παράκεντρα <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 {J_{A},J_{B},J_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A},J_{B},J_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d242305e0d47afe24fd3224e153d0ab82576d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.773ex; height:2.509ex;" alt="{\displaystyle {\rm {J_{A},J_{B},J_{\Gamma }}}}"></span> είναι σημεία εξωτερικά του τριγώνου.</li> <li>Τα 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 {\rm {A,J_{B},J_{\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> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A,J_{B},J_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24b15764095c915d9bbebe706593d75ecb90c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.856ex; height:2.509ex;" alt="{\displaystyle {\rm {A,J_{B},J_{\Gamma }}}}"></span> είναι συνευθειακά, καθώς 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 {\rm {J_{A},B,J_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">B</mi> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A},B,J_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8f4b9b165dc4a68ec864d7ee2017df56630dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.828ex; height:2.509ex;" alt="{\displaystyle {\rm {J_{A},B,J_{\Gamma }}}}"></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 {J_{A},J_{B},\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A},J_{B},\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32197e82d948debb6ada3ba949aa973e5cd4f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.771ex; height:2.509ex;" alt="{\displaystyle {\rm {J_{A},J_{B},\Gamma }}}"></span>.</li> <li>Η γωνία των εξωτερικών διχοτόμων 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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></span> είναι ίση με <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 90^{o}-{\tfrac {\hat {\rm {A}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{o}-{\tfrac {\hat {\rm {A}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab14126c42a88eded741176a63ebc5241523cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.264ex; height:4.176ex;" alt="{\displaystyle 90^{o}-{\tfrac {\hat {\rm {A}}}{2}}}"></span>.<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: 85">: 85 </span></sup></li></ul> <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</strong>   </td></tr> <tr> <td> <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 {\rm {BJ_{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">B</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BJ_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5adbf045e7ffdbb5303fd19e906d20d02e605f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.305ex; height:2.509ex;" alt="{\displaystyle {\rm {BJ_{A}}}}"></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 J_{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">Γ</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma J_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1968b12a79e53800f4985adf3d710bd550ca33fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.113ex; height:2.509ex;" alt="{\displaystyle {\rm {\Gamma J_{A}}}}"></span> είναι εξωτερικές διχοτόμοι, 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 {\angle J_{A}B\Gamma ={\tfrac {180^{o}-{\hat {B}}}{2}}\quad }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="1em" /> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle J_{A}B\Gamma ={\tfrac {180^{o}-{\hat {B}}}{2}}\quad }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837cf482384ea68aaad2c6a140da1808a9105181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.433ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle J_{A}B\Gamma ={\tfrac {180^{o}-{\hat {B}}}{2}}\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 {\rm {\angle J_{A}\Gamma B={\tfrac {180^{o}-{\hat {\Gamma }}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mi mathvariant="normal">Γ</mi> <mi mathvariant="normal">B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\rm {\angle J_{A}\Gamma B={\tfrac {180^{o}-{\hat {\Gamma }}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96eacbcd5b3a14c82b7601fe0157080ad29b494b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.297ex; height:4.176ex;" alt="{\displaystyle \quad {\rm {\angle J_{A}\Gamma B={\tfrac {180^{o}-{\hat {\Gamma }}}{2}}}}}"></span>.</dd></dl> <p>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 {B\Gamma J_{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">B</mi> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {B\Gamma J_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d631283642bbed7992ad8c61580c0625b7ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.758ex; height:2.509ex;" alt="{\displaystyle {\rm {B\Gamma J_{A}}}}"></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 {\angle BJ_{A}\Gamma =180^{o}-{\tfrac {180^{o}-{\hat {B}}}{2}}-{\tfrac {180^{o}-{\hat {\Gamma }}}{2}}={\tfrac {{\hat {B}}+{\hat {\Gamma }}}{2}}=90^{o}-{\tfrac {\hat {A}}{2}}.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle BJ_{A}\Gamma =180^{o}-{\tfrac {180^{o}-{\hat {B}}}{2}}-{\tfrac {180^{o}-{\hat {\Gamma }}}{2}}={\tfrac {{\hat {B}}+{\hat {\Gamma }}}{2}}=90^{o}-{\tfrac {\hat {A}}{2}}.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef3624c0e8dc0cb24446a228f08592c6e6112fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.21ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle BJ_{A}\Gamma =180^{o}-{\tfrac {180^{o}-{\hat {B}}}{2}}-{\tfrac {180^{o}-{\hat {\Gamma }}}{2}}={\tfrac {{\hat {B}}+{\hat {\Gamma }}}{2}}=90^{o}-{\tfrac {\hat {A}}{2}}.}}}"></span></dd></dl> </td></tr></tbody></table> <ul><li>Η γωνία της εσωτερικής διχοτόμου της <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 {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {B}}}}"></span> και της εξωτερικής διχοτόμου της <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 {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\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 {\tfrac {\hat {\rm {A}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\hat {\rm {A}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375c7fefdf2d3bfc7c25039e21230856b2581fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.069ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\hat {\rm {A}}}{2}}}"></span>.<sup id="cite_ref-Tav_1-3" 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: 85">: 85 </span></sup></li></ul> <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</strong>   </td></tr> <tr> <td> <p>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 {BJ_{A}J_{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">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BJ_{A}J_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d26350ec962551484f8301083e91a95e0e6fdd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.896ex; height:2.509ex;" alt="{\displaystyle {\rm {BJ_{A}J_{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 {\angle BJ_{A}J_{B}=180^{o}-({\tfrac {\hat {B}}{2}}+{\tfrac {180^{o}-{\hat {B}}}{2}})-(90^{o}-{\tfrac {\hat {A}}{2}})={\tfrac {\hat {A}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\angle BJ_{A}J_{B}=180^{o}-({\tfrac {\hat {B}}{2}}+{\tfrac {180^{o}-{\hat {B}}}{2}})-(90^{o}-{\tfrac {\hat {A}}{2}})={\tfrac {\hat {A}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f66fcc9c2d59f37fb4114775b4f4560d4754ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.385ex; height:4.176ex;" alt="{\displaystyle {\rm {\angle BJ_{A}J_{B}=180^{o}-({\tfrac {\hat {B}}{2}}+{\tfrac {180^{o}-{\hat {B}}}{2}})-(90^{o}-{\tfrac {\hat {A}}{2}})={\tfrac {\hat {A}}{2}}}}}"></span>.</dd></dl> </td></tr></tbody></table> <ul><li>(<b><a href="/wiki/%CE%A3%CE%B7%CE%BC%CE%B5%CE%AF%CE%BF_Gergonne" title="Σημείο Gergonne">Σημείο Gergonne</a></b>) Τα ευθύγραμμα τμήματα <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 {AI_{A}',BI_{B}',\Gamma I_{\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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{A}',BI_{B}',\Gamma I_{\Gamma }'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee59bca3d3f52113828568cba8e0e91bd559fb2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.548ex; height:2.843ex;" alt="{\displaystyle {\rm {AI_{A}',BI_{B}',\Gamma I_{\Gamma }'}}}"></span> διέρχονται από το ίδιο σημείο.<sup id="cite_ref-P74_3-2" class="reference"><a href="#cite_note-P74-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 36">: 36 </span></sup></li> <li>Ισχύει 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 {\rm {AI_{B}'=AI_{B}''=BI_{A}'=BI_{A}''=\tau -\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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">A</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{B}'=AI_{B}''=BI_{A}'=BI_{A}''=\tau -\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ceae3bb06b4c27ef7e7c77a02a0af725171f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.556ex; height:2.843ex;" alt="{\displaystyle {\rm {AI_{B}'=AI_{B}''=BI_{A}'=BI_{A}''=\tau -\gamma }}}"></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 {AI_{\Gamma }'=AI_{\Gamma }''=\Gamma I_{A}'=\Gamma I_{A}''=\tau -\beta }}}"> <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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">A</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{\Gamma }'=AI_{\Gamma }''=\Gamma I_{A}'=\Gamma I_{A}''=\tau -\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad827e30535e3d5050f1f9bc217e17ed7a9a871a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.967ex; height:2.843ex;" alt="{\displaystyle {\rm {AI_{\Gamma }'=AI_{\Gamma }''=\Gamma I_{A}'=\Gamma I_{A}''=\tau -\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 {\rm {BI_{\Gamma }'=BI_{\Gamma }''=\Gamma I_{B}'=\Gamma I_{B}''=\tau -\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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI_{\Gamma }'=BI_{\Gamma }''=\Gamma I_{B}'=\Gamma I_{B}''=\tau -\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52466e923716030c89c501c6c344797c487e496c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.564ex; height:2.843ex;" alt="{\displaystyle {\rm {BI_{\Gamma }'=BI_{\Gamma }''=\Gamma I_{B}'=\Gamma I_{B}''=\tau -\gamma }}}"></span>, όπου <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 \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4834dcbe34b65edd6abdf2b075171b33a2c68814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.21ex; height:3.509ex;" alt="{\displaystyle \tau ={\tfrac {1}{2}}\cdot (\alpha +\beta +\gamma )}"></span> η <a href="/wiki/%CE%97%CE%BC%CE%B9%CF%80%CE%B5%CF%81%CE%AF%CE%BC%CE%B5%CF%84%CF%81%CE%BF%CF%82" title="Ημιπερίμετρος">ημιπερίμετρος</a>.<sup id="cite_ref-Tav_1-4" 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: 86-87">: 86-87 </span></sup></li></ul> <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</strong>   </td></tr> <tr> <td> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Parakentro_j_a_lengths_proof_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Parakentro_j_a_lengths_proof_el.svg/220px-Parakentro_j_a_lengths_proof_el.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Parakentro_j_a_lengths_proof_el.svg/330px-Parakentro_j_a_lengths_proof_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Parakentro_j_a_lengths_proof_el.svg/440px-Parakentro_j_a_lengths_proof_el.svg.png 2x" data-file-width="335" data-file-height="151" /></a><figcaption>Σχήμα απόδειξης.</figcaption></figure> <p>Ξεκινάμε παρατηρώντας 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 {\rm {AI_{A}''=AI_{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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">A</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{A}''=AI_{A}'''}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7721f8d6f5b86d970cc7e9843c489cc01508ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.318ex; height:2.843ex;" alt="{\displaystyle {\rm {AI_{A}''=AI_{A}'''}}}"></span> ως τα <a href="/wiki/%CE%95%CF%85%CE%B8%CE%B5%CE%AF%CE%B1_%CE%B5%CF%86%CE%B1%CF%80%CF%84%CF%8C%CE%BC%CE%B5%CE%BD%CE%B7_%CF%83%CE%B5_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF" title="Ευθεία εφαπτόμενη σε κύκλο">ευθύγραμμα τμήματα 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 {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> εφαπτόμενα σε κύκλο</a>. Για τον ίδιο λόγο 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 {\rm {BI_{A}''=BI_{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">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {BI_{A}''=BI_{A}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca21db40dd35754d6a9c6da4e2aebf73d98962b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.998ex; height:2.843ex;" alt="{\displaystyle {\rm {BI_{A}''=BI_{A}'}}}"></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 I_{A}'=\Gamma I_{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">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\Gamma I_{A}'=\Gamma I_{A}'''}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7fb52bf3ba8a5cc483638605cab11af6d994a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.738ex; height:2.843ex;" alt="{\displaystyle {\rm {\Gamma I_{A}'=\Gamma I_{A}'''}}}"></span>. Επομένως, παίρνουμε το σύστημα </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 {\begin{aligned}y+\beta &=x+\gamma \\x+y&=\alpha .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo>+</mo> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y+\beta &=x+\gamma \\x+y&=\alpha .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363d8a7d8d87f8b242dea2edf5a29a94fbbdc309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.61ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}y+\beta &=x+\gamma \\x+y&=\alpha .\end{aligned}}}"></span></dd></dl> <p>Το σύστημα έχει λύση </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 x=\tau -\gamma \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\tau -\gamma \quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07946537750f7cea2537f9a2268bb7929e67143f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.055ex; height:2.509ex;" alt="{\displaystyle x=\tau -\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 y=\tau -\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad y=\tau -\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06fba00f90a13def8b241732458e0393e567e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.951ex; height:2.509ex;" alt="{\displaystyle \quad y=\tau -\beta }"></span>.</dd></dl> <p>Αντίστοιχα και για τα υπόλοιπα τμήματα. </p> </td></tr></tbody></table> <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 {\rm {A'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">A</mi> <mo>′</mo> </msup> </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/031eb4f7dbc034d5da715fc22ad212c1281e41e3" 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 {\rm {A'}}}"></span> το σημείο τομής της προέκτασης της <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 {AI}}}"> <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">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7ffeddaf0345dd0f6cd62a427cc2e54b35610d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle {\rm {AI}}}"></span> με τον <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">περιγεγραμμένο κύκλο</a>, τότε<sup id="cite_ref-Tav_1-5" 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: 85">: 85 </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 {\rm {A'B=A'I=A'\Gamma =A'I_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">A</mi> <mo>′</mo> </msup> <mi mathvariant="normal">B</mi> <mo>=</mo> <msup> <mi mathvariant="normal">A</mi> <mo>′</mo> </msup> <mi mathvariant="normal">I</mi> <mo>=</mo> <msup> <mi mathvariant="normal">A</mi> <mo>′</mo> </msup> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <msup> <mi mathvariant="normal">A</mi> <mo>′</mo> </msup> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A'B=A'I=A'\Gamma =A'I_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c87efe1c53191d40f0dce51a464f610801863ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.249ex; height:2.843ex;" alt="{\displaystyle {\rm {A'B=A'I=A'\Gamma =A'I_{A}}}}"></span>.</dd></dl></dd></dl> <ul><li>(<b><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></b>) 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 {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> είναι 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 κύκλος</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 {OJ_{A}} ^{2}=R^{2}+2R\rho _{\mathrm {A} },\quad \mathrm {OJ_{B}} ^{2}=R^{2}+2R\rho _{\mathrm {B} }\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">J</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> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>R</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>R</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {OJ_{A}} ^{2}=R^{2}+2R\rho _{\mathrm {A} },\quad \mathrm {OJ_{B}} ^{2}=R^{2}+2R\rho _{\mathrm {B} }\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42faa8996e8823803ad8c27e9e2cc56e2fd8f39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.286ex; height:3.176ex;" alt="{\displaystyle \mathrm {OJ_{A}} ^{2}=R^{2}+2R\rho _{\mathrm {A} },\quad \mathrm {OJ_{B}} ^{2}=R^{2}+2R\rho _{\mathrm {B} }\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 \mathrm {OJ_{\Gamma }} ^{2}=R^{2}+2R\rho _{\mathrm {\Gamma } }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>R</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mathrm {OJ_{\Gamma }} ^{2}=R^{2}+2R\rho _{\mathrm {\Gamma } }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b1228af10ca0b88468e1649d01513ce2f3ef3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.785ex; height:3.176ex;" alt="{\displaystyle \quad \mathrm {OJ_{\Gamma }} ^{2}=R^{2}+2R\rho _{\mathrm {\Gamma } }}"></span>.</dd></dl></dd></dl> <ul><li>Το εμβαδόν 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> δίνεται από τους τύπους:<sup id="cite_ref-P74_3-3" class="reference"><a href="#cite_note-P74-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 45">: 45 </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 {\rm {E=(\tau -\alpha )\cdot \rho _{\rm {A}}=(\tau -\beta )\cdot \rho _{\rm {B}}=(\tau -\gamma )\cdot \rho _{\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">E</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <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>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>,</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E=(\tau -\alpha )\cdot \rho _{\rm {A}}=(\tau -\beta )\cdot \rho _{\rm {B}}=(\tau -\gamma )\cdot \rho _{\rm {\Gamma }},}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94af7ddd389a52ceaa8f3fba0ceef7f63fd4ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.925ex; height:2.843ex;" alt="{\displaystyle {\rm {E=(\tau -\alpha )\cdot \rho _{\rm {A}}=(\tau -\beta )\cdot \rho _{\rm {B}}=(\tau -\gamma )\cdot \rho _{\rm {\Gamma }},}}}"></span></dd></dl></dd> <dd>και <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 {E={\sqrt {\rho \cdot \rho _{\rm {A}}\cdot \rho _{\rm {B}}\cdot \rho _{\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">E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ</mi> <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>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E={\sqrt {\rho \cdot \rho _{\rm {A}}\cdot \rho _{\rm {B}}\cdot \rho _{\rm {\Gamma }}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa4bc333559e991c8a547eedeba0d89938662d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.582ex; height:3.009ex;" alt="{\displaystyle {\rm {E={\sqrt {\rho \cdot \rho _{\rm {A}}\cdot \rho _{\rm {B}}\cdot \rho _{\rm {\Gamma }}}}}}}"></span>.</dd></dl></dd></dl> <ul><li>Από τον <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">τύπο του Ήρωνα</a>, η ακτίνα του παρεγγεγραμμένου κύκλου δίνεται από τον τύπο<sup id="cite_ref-K75_6-1" class="reference"><a href="#cite_note-K75-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 139">: 139 </span></sup><sup id="cite_ref-Tog_5-2" class="reference"><a href="#cite_note-Tog-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 127">: 127 </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 \rho _{\mathrm {A} }={\frac {\mathrm {E} }{\tau -\alpha }}={\sqrt {\frac {\tau \cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau -\alpha }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {A} }={\frac {\mathrm {E} }{\tau -\alpha }}={\sqrt {\frac {\tau \cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau -\alpha }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4eebd5bf1ee70da30969d344d7bc2528c3ce24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.894ex; height:7.676ex;" alt="{\displaystyle \rho _{\mathrm {A} }={\frac {\mathrm {E} }{\tau -\alpha }}={\sqrt {\frac {\tau \cdot (\tau -\beta )\cdot (\tau -\gamma )}{\tau -\alpha }}},}"></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 \quad \rho _{\mathrm {B} }={\frac {\mathrm {E} }{\tau -\beta }}={\sqrt {\frac {\tau \cdot (\tau -\gamma )\cdot (\tau -\alpha )}{\tau -\beta }}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\mathrm {B} }={\frac {\mathrm {E} }{\tau -\beta }}={\sqrt {\frac {\tau \cdot (\tau -\gamma )\cdot (\tau -\alpha )}{\tau -\beta }}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b9bd0dded6edbc7f47a1e1c56579f8eccd19f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.824ex; height:7.676ex;" alt="{\displaystyle \quad \rho _{\mathrm {B} }={\frac {\mathrm {E} }{\tau -\beta }}={\sqrt {\frac {\tau \cdot (\tau -\gamma )\cdot (\tau -\alpha )}{\tau -\beta }}}\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 \rho _{\mathrm {\Gamma } }={\frac {\mathrm {E} }{\tau -\gamma }}={\sqrt {\frac {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )}{\tau -\gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {\Gamma } }={\frac {\mathrm {E} }{\tau -\gamma }}={\sqrt {\frac {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )}{\tau -\gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7fa32f998968d6c8de41415935b902aee6b730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.042ex; height:7.509ex;" alt="{\displaystyle \rho _{\mathrm {\Gamma } }={\frac {\mathrm {E} }{\tau -\gamma }}={\sqrt {\frac {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )}{\tau -\gamma }}}}"></span>.</dd></dl></dd></dl> <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</strong>   </td></tr> <tr> <td> <p>Από το σχήμα 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 {E_{AB\Gamma }=E_{AJ_{A}I_{A}''}+E_{AJ_{A}I_{A}'''}-E_{BI_{A}'I_{A}''J_{A}}-E_{J_{A}I_{A}'\Gamma I_{A}''}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{AB\Gamma }=E_{AJ_{A}I_{A}''}+E_{AJ_{A}I_{A}'''}-E_{BI_{A}'I_{A}''J_{A}}-E_{J_{A}I_{A}'\Gamma I_{A}''}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4640a73e95096bf7594b678bec7f0fb8b261c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:47.467ex; height:3.176ex;" alt="{\displaystyle {\rm {E_{AB\Gamma }=E_{AJ_{A}I_{A}''}+E_{AJ_{A}I_{A}'''}-E_{BI_{A}'I_{A}''J_{A}}-E_{J_{A}I_{A}'\Gamma I_{A}''}}}}"></span>.</dd></dl> <p>Αφού 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 {\rm {J_{A}I_{A}'B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}I_{A}'B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4000bb6c6cedd1c5aa42765bdba94285833f876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.61ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{A}I_{A}'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 {J_{A}I_{A}''B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}I_{A}''B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b4b062528cbefd6fe271d23ab7d348926a4524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.61ex; height:2.843ex;" alt="{\displaystyle {\rm {J_{A}I_{A}''B}}}"></span> είναι ίσα, 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 {E_{BI_{A}'I_{A}''J_{A}}=2\cdot E_{BJ_{A}I_{A}''}}}=\rho _{\rm {A}}\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> </mrow> </msub> </mrow> </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>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{BI_{A}'I_{A}''J_{A}}=2\cdot E_{BJ_{A}I_{A}''}}}=\rho _{\rm {A}}\cdot x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b304a1d51ea7550eaabd7ee4763f648e04300701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.967ex; height:3.176ex;" alt="{\displaystyle {\rm {E_{BI_{A}'I_{A}''J_{A}}=2\cdot E_{BJ_{A}I_{A}''}}}=\rho _{\rm {A}}\cdot x}"></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 {\rm {E_{J_{A}I_{A}'\Gamma I_{A}''}=2\cdot E_{\Gamma J_{A}I_{A}'''}}}=\rho _{\rm {A}}\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>″</mo> </msubsup> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>‴</mo> </msubsup> </mrow> </msub> </mrow> </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>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E_{J_{A}I_{A}'\Gamma I_{A}''}=2\cdot E_{\Gamma J_{A}I_{A}'''}}}=\rho _{\rm {A}}\cdot y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31f3bdcc7ca564315542efd4f6ad071d5758df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.621ex; height:3.176ex;" alt="{\displaystyle {\rm {E_{J_{A}I_{A}'\Gamma I_{A}''}=2\cdot E_{\Gamma J_{A}I_{A}'''}}}=\rho _{\rm {A}}\cdot y}"></span>.</dd></dl> <p>Συνδυάζοντας τα παραπάνω και 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 x+y=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9908a4e2082ab0c1ed99212d01bd232fdc3c68c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.912ex; height:2.343ex;" alt="{\displaystyle x+y=\alpha }"></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 {\begin{aligned}{\rm {E_{AB\Gamma }}}&={\tfrac {1}{2}}\cdot (\gamma +x)\cdot \rho _{\rm {A}}+{\tfrac {1}{2}}\cdot (\beta +y)\cdot \rho _{\rm {A}}-\rho _{\rm {A}}\cdot x-\rho _{\rm {A}}\cdot y\\&=\rho _{\rm {A}}\cdot (\tau -\alpha ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <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>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>x</mi> <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>⋅</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\rm {E_{AB\Gamma }}}&={\tfrac {1}{2}}\cdot (\gamma +x)\cdot \rho _{\rm {A}}+{\tfrac {1}{2}}\cdot (\beta +y)\cdot \rho _{\rm {A}}-\rho _{\rm {A}}\cdot x-\rho _{\rm {A}}\cdot y\\&=\rho _{\rm {A}}\cdot (\tau -\alpha ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e8f1b1c6dd23d36c0ad44466d9c6c7a5913a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.532ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}{\rm {E_{AB\Gamma }}}&={\tfrac {1}{2}}\cdot (\gamma +x)\cdot \rho _{\rm {A}}+{\tfrac {1}{2}}\cdot (\beta +y)\cdot \rho _{\rm {A}}-\rho _{\rm {A}}\cdot x-\rho _{\rm {A}}\cdot y\\&=\rho _{\rm {A}}\cdot (\tau -\alpha ).\end{aligned}}}"></span></dd></dl> </td></tr></tbody></table> <ul><li>Επίσης, οι ακτίνες των παρεγγεγραμμένων κύκλων δίνονται από τις τριγωνομετρικές σχέσεις<sup id="cite_ref-P57_7-1" class="reference"><a href="#cite_note-P57-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:264</sup><sup id="cite_ref-P74_3-4" class="reference"><a href="#cite_note-P74-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 46-47">: 46-47 </span></sup><sup id="cite_ref-Tog_5-3" class="reference"><a href="#cite_note-Tog-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 127">: 127 </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 \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\rm {B}}{2}}\cdot \cos {\frac {\rm {\Gamma }}{2}}}{\cos {\frac {\rm {A}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\rm {B}}{2}}\cdot \cos {\frac {\rm {\Gamma }}{2}}}{\cos {\frac {\rm {A}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad14db3b2858646f13d8ad5b52377fe3302e304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.307ex; height:8.009ex;" alt="{\displaystyle \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\rm {B}}{2}}\cdot \cos {\frac {\rm {\Gamma }}{2}}}{\cos {\frac {\rm {A}}{2}}}}}"></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 \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\Gamma }{2}}\cdot \cos {\frac {\rm {A}}{2}}}{\cos {\frac {\rm {B}}{2}}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\Gamma }{2}}\cdot \cos {\frac {\rm {A}}{2}}}{\cos {\frac {\rm {B}}{2}}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d46b24e16131333bc6dec8d637e8d0f6f172d8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:26.796ex; height:8.176ex;" alt="{\displaystyle \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\Gamma }{2}}\cdot \cos {\frac {\rm {A}}{2}}}{\cos {\frac {\rm {B}}{2}}}}\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 \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\rm {A}}{2}}\cdot \cos {\frac {\rm {B}}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\rm {A}}{2}}\cdot \cos {\frac {\rm {B}}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8aeb9bdd071356f5cc5dadf2b8e641386b1e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.404ex; height:8.009ex;" alt="{\displaystyle \quad \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\rm {A}}{2}}\cdot \cos {\frac {\rm {B}}{2}}}{\cos {\frac {\rm {\Gamma }}{2}}}}}"></span>,</dd></dl></dd> <dd>και επίσης <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 \rho _{\rm {A}}=\tau \cdot \tan {\frac {\rm {A}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {A}}=\tau \cdot \tan {\frac {\rm {A}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb72c15400d2db1375b43f8bd2da6c004fddd88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.972ex; height:5.343ex;" alt="{\displaystyle \rho _{\rm {A}}=\tau \cdot \tan {\frac {\rm {A}}{2}}}"></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 \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\rm {B}}{2}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\rm {B}}{2}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65bdfea602e771c3e59d4064410c50dcdd5facad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.451ex; height:5.176ex;" alt="{\displaystyle \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\rm {B}}{2}}\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 \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\Gamma }}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\Gamma }}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a2373502b6fdf0013afe368cb0a0506cca90315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.799ex; height:5.176ex;" alt="{\displaystyle \quad \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\Gamma }}{2}}}"></span>.</dd></dl></dd></dl> <ul><li>(<b><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></b>) 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 {\rm {I_{A}',I_{B}',I_{\Gamma }'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {I_{A}',I_{B}',I_{\Gamma }'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76081ed9d78763087f8c1456689ae68a125fa070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.707ex; height:2.843ex;" alt="{\displaystyle {\rm {I_{A}',I_{B}',I_{\Gamma }'}}}"></span> τα σημεία επαφής των παρεγγεγραμμένων κύκλων με κέντρα <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 {J_{A},J_{B},J_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A},J_{B},J_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d242305e0d47afe24fd3224e153d0ab82576d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.773ex; height:2.509ex;" alt="{\displaystyle {\rm {J_{A},J_{B},J_{\Gamma }}}}"></span> με τις πλευρές <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 ,B\Gamma ,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">Γ</mi> <mo>,</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {A\Gamma ,B\Gamma ,AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec21efe83403e027972ec5c532a5592b77a6fcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.751ex; height:2.509ex;" alt="{\displaystyle {\rm {A\Gamma ,B\Gamma ,AB}}}"></span> του τριγώνου, τότε 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 {AI_{A}',BI_{B}',\Gamma I_{\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> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mi mathvariant="normal">B</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mi mathvariant="normal">Γ</mi> <msubsup> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AI_{A}',BI_{B}',\Gamma I_{\Gamma }'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee59bca3d3f52113828568cba8e0e91bd559fb2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.548ex; height:2.843ex;" alt="{\displaystyle {\rm {AI_{A}',BI_{B}',\Gamma I_{\Gamma }'}}}"></span> συντρέχουν στο σημείο Νάγκελ.</li> <li>Οι εσωτερικές διχοτόμοι 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> είναι <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 του τριγώνου</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 {\rm {J_{A}J_{B}J_{\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </msub> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </msub> <msub> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {J_{A}J_{B}J_{\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cb09cf2a77ceb4899470d9097d868701e11bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.705ex; height:2.509ex;" alt="{\displaystyle {\rm {J_{A}J_{B}J_{\Gamma }}}}"></span>.</li> <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 περιγεγραμμένου κύκλου, τότε ισχύει 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 \rho _{\rm {A}}+\rho _{\rm {B}}+\rho _{\rm {\Gamma }}=\rho +4R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mo>+</mo> <mn>4</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {A}}+\rho _{\rm {B}}+\rho _{\rm {\Gamma }}=\rho +4R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2982c8e4b646063a19bdf77accabf81d52d9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.676ex;" alt="{\displaystyle \rho _{\rm {A}}+\rho _{\rm {B}}+\rho _{\rm {\Gamma }}=\rho +4R}"></span>.<sup id="cite_ref-Tav_1-6" 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: 87">: 87 </span></sup></li> <li>Οι τριγραμμικές συντεταγμένες των παρακέντρων 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 -1:1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1:1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2094c225f92b11b3226f6b2fd84934a7e23f35e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.17ex; height:2.343ex;" alt="{\displaystyle -1:1:1}"></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 1:-1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:-1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9464c7e61f2a87bd2ae8ac803ec0490d04be731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.17ex; height:2.343ex;" alt="{\displaystyle 1:-1:1}"></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 1:1:-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1:-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0596fb532981a4774928b32d14de0389f439b17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.17ex; height:2.343ex;" alt="{\displaystyle 1:1:-1}"></span> αντίστοιχα.</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%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%CE%B1%CE%B9_%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&veaction=edit&section=7" 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%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%CE%B1%CE%B9_%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&action=edit&section=7" 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%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">Διχοτόμος γωνίας</a></li> <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" class="mw-redirect" title="Περιγεγραμμένος κύκλος">Περιγεγραμμένος κύκλος</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></li> <li><a href="/wiki/%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="Ορθόκεντρο τριγώνου">Ορθόκεντρο τριγώνου</a></li></ul> <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%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%CE%B1%CE%B9_%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&veaction=edit&section=8" 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%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%CE%B1%CE%B9_%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&action=edit&section=8" 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> <sup><a href="#cite_ref-Tav_1-3">1,3</a></sup> <sup><a href="#cite_ref-Tav_1-4">1,4</a></sup> <sup><a href="#cite_ref-Tav_1-5">1,5</a></sup> <sup><a href="#cite_ref-Tav_1-6">1,6</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%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-T57-2"><span class="mw-cite-backlink"><a href="#cite_ref-T57_2-0">↑</a></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%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-P74-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-P74_3-0">3,0</a></sup> <sup><a href="#cite_ref-P74_3-1">3,1</a></sup> <sup><a href="#cite_ref-P74_3-2">3,2</a></sup> <sup><a href="#cite_ref-P74_3-3">3,3</a></sup> <sup><a href="#cite_ref-P74_3-4">3,4</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%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book">Στεργίου, Χαράλαμπος (2011). <i>Γεωμετρία για διαγωνισμούς: Τρίγωνα, τετράπλευρα, κύκλος, εγγράψιμα</i>. Αθήνα: Σαββάλας. <a href="/wiki/%CE%94%CE%B9%CE%B5%CE%B8%CE%BD%CE%AE%CF%82_%CF%80%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%B2%CE%B9%CE%B2%CE%BB%CE%AF%CE%BF%CF%85" title="Διεθνής πρότυπος αριθμός βιβλίου">ISBN</a> <a href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%A0%CE%B7%CE%B3%CE%AD%CF%82%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CF%89%CE%BD/9789604930357" title="Ειδικό:ΠηγέςΒιβλίων/9789604930357">9789604930357</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1+%CE%B3%CE%B9%CE%B1+%CE%B4%CE%B9%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CF%83%CE%BC%CE%BF%CF%8D%CF%82%3A+%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%B1%2C+%CF%84%CE%B5%CF%84%CF%81%CE%AC%CF%80%CE%BB%CE%B5%CF%85%CF%81%CE%B1%2C+%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82%2C+%CE%B5%CE%B3%CE%B3%CF%81%CE%AC%CF%88%CE%B9%CE%BC%CE%B1&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%A3%CE%B1%CE%B2%CE%B2%CE%AC%CE%BB%CE%B1%CF%82&rft.date=2011&rft.isbn=9789604930357&rft.aulast=%CE%A3%CF%84%CE%B5%CF%81%CE%B3%CE%AF%CE%BF%CF%85&rft.aufirst=%CE%A7%CE%B1%CF%81%CE%AC%CE%BB%CE%B1%CE%BC%CF%80%CE%BF%CF%82&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Tog-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Tog_5-0">5,0</a></sup> <sup><a href="#cite_ref-Tog_5-1">5,1</a></sup> <sup><a href="#cite_ref-Tog_5-2">5,2</a></sup> <sup><a href="#cite_ref-Tog_5-3">5,3</a></sup></span> <span class="reference-text"><cite class="citation book">Τόγκας, Πέτρος Γ. <i>Ασκήσεις και προβλήματα τριγωνομετρίας</i>. Αθήνα: Εκδοτικός οίκος Πέτρου Γ. Τόγκα Ο.Ε.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%91%CF%83%CE%BA%CE%AE%CF%83%CE%B5%CE%B9%CF%82+%CE%BA%CE%B1%CE%B9+%CF%80%CF%81%CE%BF%CE%B2%CE%BB%CE%AE%CE%BC%CE%B1%CF%84%CE%B1+%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1%CF%82&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%95%CE%BA%CE%B4%CE%BF%CF%84%CE%B9%CE%BA%CF%8C%CF%82+%CE%BF%CE%AF%CE%BA%CE%BF%CF%82+%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%85+%CE%93.+%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1+%CE%9F.%CE%95&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%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-K75-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-K75_6-0">6,0</a></sup> <sup><a href="#cite_ref-K75_6-1">6,1</a></sup></span> <span class="reference-text"><cite class="citation book">Κανέλλος, Σπ. Γ. (1975). <i>Ευκλείδειος Γεωμετρία</i>. Αθήνα 1975: Οργανισμός Εκδόσεων 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%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%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+1975&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%CE%BD+%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=1975&rft.aulast=%CE%9A%CE%B1%CE%BD%CE%AD%CE%BB%CE%BB%CE%BF%CF%82&rft.aufirst=%CE%A3%CF%80.+%CE%93.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-P57-7"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-P57_7-0">7,0</a></sup> <sup><a href="#cite_ref-P57_7-1">7,1</a></sup></span> <span class="reference-text"><cite class="citation book">Τόγκας, Πέτρος Γ. (1957). <i>Ευθύγραμμος τριγωνομετρία</i>. Αθήνα: Εκδοτικός Οίκος 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%95%CF%85%CE%B8%CF%8D%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF%CF%82+%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%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%95%CE%BA%CE%B4%CE%BF%CF%84%CE%B9%CE%BA%CF%8C%CF%82+%CE%9F%CE%AF%CE%BA%CE%BF%CF%82+%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%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%CE%B1%CE%B9+%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="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> 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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></li> <li><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></li> <li><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></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" 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 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τριγώνου">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 href="/wiki/%CE%9F%CF%81%CE%B8%CE%B9%CE%BA%CF%8C_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ορθικό τρίγωνο">ορθικό</a></li> <li><a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF_Gergonne" title="Τρίγωνο Gergonne">τρίγωνο Gergonne</a></li> <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">τρίγωνο Μόρλεϊ</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span typeof="mw:File"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Portal-puzzle.svg" class="mw-file-description"><img alt="Portal icon" 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