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class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Vzhled"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Změnit vzhled velikosti písma, šířky stránky a barvy" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Vzhled" > <label id="vector-appearance-dropdown-label" for="vector-appearance-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Vzhled</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_cs.wikipedia.org&uselang=cs" class=""><span>Podpořte Wikipedii</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=Speci%C3%A1ln%C3%AD:Vytvo%C5%99it_%C3%BA%C4%8Det&returnto=Troj%C3%BAheln%C3%ADk" title="Doporučujeme vytvořit si účet a přihlásit se, ovšem není to povinné" class=""><span>Vytvoření účtu</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=Speci%C3%A1ln%C3%AD:P%C5%99ihl%C3%A1sit&returnto=Troj%C3%BAheln%C3%ADk" title="Doporučujeme vám přihlásit se, ovšem není to povinné. [o]" accesskey="o" class=""><span>Přihlášení</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="Další možnosti" > <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="Osobní nástroje" > <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">Osobní nástroje</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Uživatelské menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_cs.wikipedia.org&uselang=cs"><span>Podpořte Wikipedii</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:Vytvo%C5%99it_%C3%BA%C4%8Det&returnto=Troj%C3%BAheln%C3%ADk" title="Doporučujeme vytvořit si účet a přihlásit se, ovšem není to povinné"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Vytvoření účtu</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:P%C5%99ihl%C3%A1sit&returnto=Troj%C3%BAheln%C3%ADk" title="Doporučujeme vám přihlásit se, ovšem není to povinné. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Přihlášení</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"> Stránky pro odhlášené editory <a href="/wiki/N%C3%A1pov%C4%9Bda:%C3%9Avod" aria-label="Více informací o editování"><span>dozvědět se více</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/Speci%C3%A1ln%C3%AD:Moje_p%C5%99%C3%ADsp%C4%9Bvky" title="Seznam editací provedených z této IP adresy [y]" accesskey="y"><span>Příspěvky</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Moje_diskuse" title="Diskuse o editacích provedených z této IP adresy [n]" accesskey="n"><span>Diskuse</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> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Obsah" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container 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">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</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">(úvod)</div> </a> </li> <li id="toc-Základní_pojmy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Základní_pojmy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Základní pojmy</span> </div> </a> <button aria-controls="toc-Základní_pojmy-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>Přepnout podsekci Základní pojmy</span> </button> <ul id="toc-Základní_pojmy-sublist" class="vector-toc-list"> <li id="toc-Druhy_trojúhelníků" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Druhy_trojúhelníků"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Druhy trojúhelníků</span> </div> </a> <ul id="toc-Druhy_trojúhelníků-sublist" class="vector-toc-list"> <li id="toc-Podle_stran" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Podle_stran"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Podle stran</span> </div> </a> <ul id="toc-Podle_stran-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Podle_úhlů" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Podle_úhlů"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Podle úhlů</span> </div> </a> <ul id="toc-Podle_úhlů-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Obvod_a_obsah" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Obvod_a_obsah"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Obvod a obsah</span> </div> </a> <ul id="toc-Obvod_a_obsah-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Věty_o_trojúhelníku" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Věty_o_trojúhelníku"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Věty o trojúhelníku</span> </div> </a> <ul id="toc-Věty_o_trojúhelníku-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti_trojúhelníku" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vlastnosti_trojúhelníku"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vlastnosti trojúhelníku</span> </div> </a> <button aria-controls="toc-Vlastnosti_trojúhelníku-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>Přepnout podsekci Vlastnosti trojúhelníku</span> </button> <ul id="toc-Vlastnosti_trojúhelníku-sublist" class="vector-toc-list"> <li id="toc-Trojúhelníková_nerovnost" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trojúhelníková_nerovnost"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Trojúhelníková nerovnost</span> </div> </a> <ul id="toc-Trojúhelníková_nerovnost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Součet_úhlů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Součet_úhlů"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Součet úhlů</span> </div> </a> <ul id="toc-Součet_úhlů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Souměrnost" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Souměrnost"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Souměrnost</span> </div> </a> <ul id="toc-Souměrnost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Výška" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Výška"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Výška</span> </div> </a> <ul id="toc-Výška-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Těžnice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Těžnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Těžnice</span> </div> </a> <ul id="toc-Těžnice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Střední_příčka" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Střední_příčka"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Střední příčka</span> </div> </a> <ul id="toc-Střední_příčka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symediána" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symediána"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Symediána</span> </div> </a> <ul id="toc-Symediána-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Osy_úhlů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Osy_úhlů"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Osy úhlů</span> </div> </a> <ul id="toc-Osy_úhlů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Třetiny_úhlů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Třetiny_úhlů"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Třetiny úhlů</span> </div> </a> <ul id="toc-Třetiny_úhlů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Osy_stran" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Osy_stran"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.10</span> <span>Osy stran</span> </div> </a> <ul id="toc-Osy_stran-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eulerova_přímka" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eulerova_přímka"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.11</span> <span>Eulerova přímka</span> </div> </a> <ul id="toc-Eulerova_přímka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussova_přímka" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gaussova_přímka"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.12</span> <span>Gaussova přímka</span> </div> </a> <ul id="toc-Gaussova_přímka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kružnice_opsaná" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kružnice_opsaná"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.13</span> <span>Kružnice opsaná</span> </div> </a> <ul id="toc-Kružnice_opsaná-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kružnice_vepsaná" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kružnice_vepsaná"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.14</span> <span>Kružnice vepsaná</span> </div> </a> <ul id="toc-Kružnice_vepsaná-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kružnice_připsaná" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kružnice_připsaná"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.15</span> <span>Kružnice připsaná</span> </div> </a> <ul id="toc-Kružnice_připsaná-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kružnice_devíti_bodů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kružnice_devíti_bodů"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.16</span> <span>Kružnice devíti bodů</span> </div> </a> <ul id="toc-Kružnice_devíti_bodů-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Konstrukce_trojúhelníku" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Konstrukce_trojúhelníku"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Konstrukce trojúhelníku</span> </div> </a> <ul id="toc-Konstrukce_trojúhelníku-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odkazy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Odkazy</span> </div> </a> <button aria-controls="toc-Odkazy-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>Přepnout podsekci Odkazy</span> </button> <ul id="toc-Odkazy-sublist" class="vector-toc-list"> <li id="toc-Literatura" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Literatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Literatura</span> </div> </a> <ul id="toc-Literatura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-sublist" class="vector-toc-list"> </ul> </li> </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="Obsah" 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="Přepnout obsah" > <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">Přepnout obsah</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">Trojúhelník</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="Přejděte k článku v jiném jazyce. Je dostupný v 162 jazycích" > <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-162" 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">162 jazyků</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ab mw-list-item"><a href="https://ab.wikipedia.org/wiki/%D0%90%D1%85%D0%BA%D3%99%D0%B0%D0%BA%D1%8C" title="Ахкәакь – abcházština" lang="ab" hreflang="ab" data-title="Ахкәакь" data-language-autonym="Аԥсшәа" data-language-local-name="abcházština" class="interlanguage-link-target"><span>Аԥсшәа</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Driehoek" title="Driehoek – afrikánština" lang="af" hreflang="af" data-title="Driehoek" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Dreieck" title="Dreieck – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Dreieck" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%B6%E1%88%B5%E1%89%B5_%E1%88%9B%E1%8A%A5%E1%8B%98%E1%8A%95" title="ሶስት ማእዘን – amharština" lang="am" hreflang="am" data-title="ሶስት ማእዘን" data-language-autonym="አማርኛ" data-language-local-name="amharština" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Trianglo" title="Trianglo – aragonština" lang="an" hreflang="an" data-title="Trianglo" data-language-autonym="Aragonés" data-language-local-name="aragonština" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/%C3%9Er%C4%ABecge" title="Þrīecge – staroangličtina" lang="ang" hreflang="ang" data-title="Þrīecge" data-language-autonym="Ænglisc" data-language-local-name="staroangličtina" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – arabština" lang="ar" hreflang="ar" data-title="مثلث" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%A1%DC%AC%DC%A0%DC%AC%DC%90" title="ܡܬܠܬܐ – aramejština" lang="arc" hreflang="arc" data-title="ܡܬܠܬܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="aramejština" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%AA%D9%84%D8%AA" title="متلت – arabština (marocká)" lang="ary" hreflang="ary" data-title="متلت" data-language-autonym="الدارجة" data-language-local-name="arabština (marocká)" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – arabština (egyptská)" lang="arz" hreflang="arz" data-title="مثلث" data-language-autonym="مصرى" data-language-local-name="arabština (egyptská)" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A7%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C" title="ত্ৰিভুজ – ásámština" lang="as" hreflang="as" data-title="ত্ৰিভুজ" data-language-autonym="অসমীয়া" data-language-local-name="ásámština" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tri%C3%A1ngulu" title="Triángulu – asturština" lang="ast" hreflang="ast" data-title="Triángulu" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Mujina" title="Mujina – ajmarština" lang="ay" hreflang="ay" data-title="Mujina" data-language-autonym="Aymar aru" data-language-local-name="ajmarština" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%9C%C3%A7bucaq" title="Üçbucaq – ázerbájdžánština" lang="az" hreflang="az" data-title="Üçbucaq" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%88%DA%86%E2%80%8C%D8%A8%D9%88%D8%AC%D8%A7%D9%82" title="اوچ‌بوجاق – South Azerbaijani" lang="azb" hreflang="azb" data-title="اوچ‌بوجاق" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D3%A8%D1%81%D0%BC%D3%A9%D0%B9%D3%A9%D1%88" title="Өсмөйөш – baškirština" lang="ba" hreflang="ba" data-title="Өсмөйөш" data-language-autonym="Башҡортса" data-language-local-name="baškirština" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Dreieck" title="Dreieck – bavorština" lang="bar" hreflang="bar" data-title="Dreieck" data-language-autonym="Boarisch" data-language-local-name="bavorština" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Tr%C4%97kompis" title="Trėkompis – žemaitština" lang="sgs" hreflang="sgs" data-title="Trėkompis" data-language-autonym="Žemaitėška" data-language-local-name="žemaitština" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Trianggulo" title="Trianggulo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Trianggulo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA" title="Трохвугольнік – běloruština" lang="be" hreflang="be" data-title="Трохвугольнік" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%BA%D1%83%D1%82%D0%BD%D1%96%D0%BA" title="Трыкутнік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Трыкутнік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA" title="Триъгълник – bulharština" lang="bg" hreflang="bg" data-title="Триъгълник" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – Bhojpuri" lang="bh" hreflang="bh" data-title="त्रिभुज" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C" title="ত্রিভুজ – bengálština" lang="bn" hreflang="bn" data-title="ত্রিভুজ" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BD%9F%E0%BD%B4%E0%BD%A2%E0%BC%8B%E0%BD%82%E0%BD%A6%E0%BD%B4%E0%BD%98%E0%BC%8B%E0%BD%91%E0%BD%96%E0%BD%96%E0%BE%B1%E0%BD%B2%E0%BC%8D" title="སཟུར་གསུམ་དབབྱི། – tibetština" lang="bo" hreflang="bo" data-title="སཟུར་གསུམ་དབབྱི།" data-language-autonym="བོད་ཡིག" data-language-local-name="tibetština" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Tric%27horn" title="Tric'horn – bretonština" lang="br" hreflang="br" data-title="Tric'horn" data-language-autonym="Brezhoneg" data-language-local-name="bretonština" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trougao" title="Trougao – bosenština" lang="bs" hreflang="bs" data-title="Trougao" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Triangle" title="Triangle – katalánština" lang="ca" hreflang="ca" data-title="Triangle" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/S%C4%83ng-g%C3%A1e%CC%A4k-h%C3%ACng" title="Săng-gáe̤k-hìng – Mindong" lang="cdo" hreflang="cdo" data-title="Săng-gáe̤k-hìng" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%A6%E1%8E%A2_%E1%8F%A7%E1%8F%85%E1%8F%8F%E1%8F%AF_%E1%8E%A4%E1%8F%83%E1%8F%B4%E1%8E%A9" title="ᏦᎢ ᏧᏅᏏᏯ ᎤᏃᏴᎩ – čerokézština" lang="chr" hreflang="chr" data-title="ᏦᎢ ᏧᏅᏏᏯ ᎤᏃᏴᎩ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="čerokézština" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95" title="سێگۆشە – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="سێگۆشە" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Triangulu" title="Triangulu – korsičtina" lang="co" hreflang="co" data-title="Triangulu" data-language-autonym="Corsu" data-language-local-name="korsičtina" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Trz%C3%ABn%C3%B3rt" title="Trzënórt – kašubština" lang="csb" hreflang="csb" data-title="Trzënórt" data-language-autonym="Kaszëbsczi" data-language-local-name="kašubština" class="interlanguage-link-target"><span>Kaszëbsczi</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" title="Виçкĕтеслĕх – čuvaština" lang="cv" hreflang="cv" data-title="Виçкĕтеслĕх" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" 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/Triongl" title="Triongl – velština" lang="cy" hreflang="cy" data-title="Triongl" data-language-autonym="Cymraeg" data-language-local-name="velština" 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/Trekant" title="Trekant – dánština" lang="da" hreflang="da" data-title="Trekant" data-language-autonym="Dansk" data-language-local-name="dánština" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dreieck" title="Dreieck – němčina" lang="de" hreflang="de" data-title="Dreieck" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Hir%C3%AAk%C4%B1nari" title="Hirêkınari – Zazaki" lang="diq" hreflang="diq" data-title="Hirêkınari" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/T%C5%9Biro%C5%BEk" title="Tśirožk – dolnolužická srbština" lang="dsb" hreflang="dsb" data-title="Tśirožk" data-language-autonym="Dolnoserbski" data-language-local-name="dolnolužická srbština" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο – řečtina" lang="el" hreflang="el" data-title="Τρίγωνο" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Triangle" title="Triangle – angličtina" lang="en" hreflang="en" data-title="Triangle" data-language-autonym="English" data-language-local-name="angličtina" 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/Triangulo" title="Triangulo – esperanto" lang="eo" hreflang="eo" data-title="Triangulo" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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/Tri%C3%A1ngulo" title="Triángulo – španělština" lang="es" hreflang="es" data-title="Triángulo" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kolmnurk" title="Kolmnurk – estonština" lang="et" hreflang="et" data-title="Kolmnurk" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Triangelu" title="Triangelu – baskičtina" lang="eu" hreflang="eu" data-title="Triangelu" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – perština" lang="fa" hreflang="fa" data-title="مثلث" data-language-autonym="فارسی" data-language-local-name="perština" 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/Kolmio" title="Kolmio – finština" lang="fi" hreflang="fi" data-title="Kolmio" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kolmnukk" title="Kolmnukk – võruština" lang="vro" hreflang="vro" data-title="Kolmnukk" data-language-autonym="Võro" data-language-local-name="võruština" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tututolu" title="Tututolu – fidžijština" lang="fj" hreflang="fj" data-title="Tututolu" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fidžijština" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Tr%C3%ADkantur" title="Tríkantur – faerština" lang="fo" hreflang="fo" data-title="Tríkantur" data-language-autonym="Føroyskt" data-language-local-name="faerština" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Triangle" title="Triangle – francouzština" lang="fr" hreflang="fr" data-title="Triangle" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Triihuk" title="Triihuk – fríština (severní)" lang="frr" hreflang="frr" data-title="Triihuk" data-language-autonym="Nordfriisk" data-language-local-name="fríština (severní)" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Triant%C3%A1n_(c%C3%A9imseata)" title="Triantán (céimseata) – irština" lang="ga" hreflang="ga" data-title="Triantán (céimseata)" data-language-autonym="Gaeilge" data-language-local-name="irština" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – čínština (dialekty Gan)" lang="gan" hreflang="gan" data-title="三角形" data-language-autonym="贛語" data-language-local-name="čínština (dialekty Gan)" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Triyang" title="Triyang – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Triyang" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tri%C3%A1ngulo" title="Triángulo – galicijština" lang="gl" hreflang="gl" data-title="Triángulo" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%BF%E0%AA%95%E0%AB%8B%E0%AA%A3" title="ત્રિકોણ – gudžarátština" lang="gu" hreflang="gu" data-title="ત્રિકોણ" data-language-autonym="ગુજરાતી" data-language-local-name="gudžarátština" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-guc mw-list-item"><a href="https://guc.wikipedia.org/wiki/Ap%C3%BCn%C3%BCinsheke%27einr%C3%BC" title="Apünüinsheke'einrü – wayúuština" lang="guc" hreflang="guc" data-title="Apünüinsheke'einrü" data-language-autonym="Wayuunaiki" data-language-local-name="wayúuština" class="interlanguage-link-target"><span>Wayuunaiki</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Troorane" title="Troorane – manština" lang="gv" hreflang="gv" data-title="Troorane" data-language-autonym="Gaelg" data-language-local-name="manština" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/S%C3%A2m-kok-h%C3%ACn" title="Sâm-kok-hìn – čínština (dialekty Hakka)" lang="hak" hreflang="hak" data-title="Sâm-kok-hìn" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="čínština (dialekty Hakka)" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="משולש – hebrejština" lang="he" hreflang="he" data-title="משולש" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – hindština" lang="hi" hreflang="hi" data-title="त्रिभुज" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trokut" title="Trokut – chorvatština" lang="hr" hreflang="hr" data-title="Trokut" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/T%C5%99ir%C3%B3%C5%BEk" title="Třiróžk – hornolužická srbština" lang="hsb" hreflang="hsb" data-title="Třiróžk" data-language-autonym="Hornjoserbsce" data-language-local-name="hornolužická srbština" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Triyang" title="Triyang – haitština" lang="ht" hreflang="ht" data-title="Triyang" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitština" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6g" title="Háromszög – maďarština" lang="hu" hreflang="hu" data-title="Háromszög" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Եռանկյուն – arménština" lang="hy" hreflang="hy" data-title="Եռանկյուն" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Triangulo" title="Triangulo – interlingua" lang="ia" hreflang="ia" data-title="Triangulo" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Segitiga" title="Segitiga – indonéština" lang="id" hreflang="id" data-title="Segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Triangulo" title="Triangulo – ido" lang="io" hreflang="io" data-title="Triangulo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Er%C3%ADhyrningur" title="Þríhyrningur – islandština" lang="is" hreflang="is" data-title="Þríhyrningur" data-language-autonym="Íslenska" data-language-local-name="islandština" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Triangolo" title="Triangolo – italština" lang="it" hreflang="it" data-title="Triangolo" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – japonština" lang="ja" hreflang="ja" data-title="三角形" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Chrayanggl" title="Chrayanggl – jamajská kreolština" lang="jam" hreflang="jam" data-title="Chrayanggl" data-language-autonym="Patois" data-language-local-name="jamajská kreolština" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pasagi_telu" title="Pasagi telu – javánština" lang="jv" hreflang="jv" data-title="Pasagi telu" data-language-autonym="Jawa" data-language-local-name="javánština" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%90%E1%83%9B%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94%E1%83%93%E1%83%98" title="სამკუთხედი – gruzínština" lang="ka" hreflang="ka" data-title="სამკუთხედი" data-language-autonym="ქართული" data-language-local-name="gruzínština" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/%C3%9Ashm%C3%BAyeshlik" title="Úshmúyeshlik – karakalpačtina" lang="kaa" hreflang="kaa" data-title="Úshmúyeshlik" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpačtina" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%A9%D0%B8%D0%BC%D1%8D" title="Щимэ – kabardinština" lang="kbd" hreflang="kbd" data-title="Щимэ" data-language-autonym="Адыгэбзэ" data-language-local-name="kabardinština" class="interlanguage-link-target"><span>Адыгэбзэ</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" title="Үшбұрыш – kazaština" lang="kk" hreflang="kk" data-title="Үшбұрыш" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E" title="ត្រីកោណ – khmérština" lang="km" hreflang="km" data-title="ត្រីកោណ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmérština" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B3%8B%E0%B2%A8" title="ತ್ರಿಕೋನ – kannadština" lang="kn" hreflang="kn" data-title="ತ್ರಿಕೋನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannadština" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81%ED%98%95" title="삼각형 – korejština" lang="ko" hreflang="ko" data-title="삼각형" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/S%C3%AAgo%C5%9Fe" title="Sêgoşe – kurdština" lang="ku" hreflang="ku" data-title="Sêgoşe" data-language-autonym="Kurdî" data-language-local-name="kurdština" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Trihorn" title="Trihorn – kornština" lang="kw" hreflang="kw" data-title="Trihorn" data-language-autonym="Kernowek" data-language-local-name="kornština" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D2%AE%D1%87_%D0%B1%D1%83%D1%80%D1%87%D1%82%D1%83%D0%BA" title="Үч бурчтук – kyrgyzština" lang="ky" hreflang="ky" data-title="Үч бурчтук" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Triangulum" title="Triangulum – latina" lang="la" hreflang="la" data-title="Triangulum" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Triangulo" title="Triangulo – lingua franca nova" lang="lfn" hreflang="lfn" data-title="Triangulo" data-language-autonym="Lingua Franca Nova" data-language-local-name="lingua franca nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Driehook" title="Driehook – limburština" lang="li" hreflang="li" data-title="Driehook" data-language-autonym="Limburgs" data-language-local-name="limburština" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Triangolo" title="Triangolo – ligurština" lang="lij" hreflang="lij" data-title="Triangolo" data-language-autonym="Ligure" data-language-local-name="ligurština" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Tri%C3%A0ngol" title="Triàngol – lombardština" lang="lmo" hreflang="lmo" data-title="Triàngol" data-language-autonym="Lombard" data-language-local-name="lombardština" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Mpanzi-mis%C3%A1to" title="Mpanzi-misáto – lingalština" lang="ln" hreflang="ln" data-title="Mpanzi-misáto" data-language-autonym="Lingála" data-language-local-name="lingalština" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AE%E0%BA%B9%E0%BA%9A%E0%BA%AA%E0%BA%B2%E0%BA%A1%E0%BB%81%E0%BA%88" title="ຮູບສາມແຈ – laoština" lang="lo" hreflang="lo" data-title="ຮູບສາມແຈ" data-language-autonym="ລາວ" data-language-local-name="laoština" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trikampis" title="Trikampis – litevština" lang="lt" hreflang="lt" data-title="Trikampis" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trijst%C5%ABris" title="Trijstūris – lotyština" lang="lv" hreflang="lv" data-title="Trijstūris" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Telolafy" title="Telolafy – malgaština" lang="mg" hreflang="mg" data-title="Telolafy" data-language-autonym="Malagasy" data-language-local-name="malgaština" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D0%BB%D1%83%D0%BA" title="Кумлук – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Кумлук" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Sagitigo" title="Sagitigo – minangkabau" lang="min" hreflang="min" data-title="Sagitigo" data-language-autonym="Minangkabau" data-language-local-name="minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B0%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Триаголник – makedonština" lang="mk" hreflang="mk" data-title="Триаголник" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B4%A3%E0%B4%82" title="ത്രികോണം – malajálamština" lang="ml" hreflang="ml" data-title="ത്രികോണം" data-language-autonym="മലയാളം" data-language-local-name="malajálamština" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://mn.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D0%B2%D0%B0%D0%BB%D0%B6%D0%B8%D0%BD" title="Гурвалжин – mongolština" lang="mn" hreflang="mn" data-title="Гурвалжин" data-language-autonym="Монгол" data-language-local-name="mongolština" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3" title="त्रिकोण – maráthština" lang="mr" hreflang="mr" data-title="त्रिकोण" data-language-autonym="मराठी" data-language-local-name="maráthština" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Segi_tiga" title="Segi tiga – malajština" lang="ms" hreflang="ms" data-title="Segi tiga" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Trijangolu" title="Trijangolu – maltština" lang="mt" hreflang="mt" data-title="Trijangolu" data-language-autonym="Malti" data-language-local-name="maltština" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%BC%E1%80%AD%E1%80%82%E1%80%B6" title="တြိဂံ – barmština" lang="my" hreflang="my" data-title="တြိဂံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="barmština" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – nepálština" lang="ne" hreflang="ne" data-title="त्रिभुज" data-language-autonym="नेपाली" data-language-local-name="nepálština" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82" title="स्वकुं – névárština" lang="new" hreflang="new" data-title="स्वकुं" data-language-autonym="नेपाल भाषा" data-language-local-name="névárština" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Driehoek_(meetkunde)" title="Driehoek (meetkunde) – nizozemština" lang="nl" hreflang="nl" data-title="Driehoek (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trekant" title="Trekant – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Trekant" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trekant" title="Trekant – norština (bokmål)" lang="nb" hreflang="nb" data-title="Trekant" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Trian" title="Trian – Norman" lang="nrf" hreflang="nrf" data-title="Trian" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target"><span>Nouormand</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Triangle" title="Triangle – okcitánština" lang="oc" hreflang="oc" data-title="Triangle" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%A4%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%AD%E0%AD%81%E0%AC%9C" title="ତ୍ରିଭୁଜ – urijština" lang="or" hreflang="or" data-title="ତ୍ରିଭୁଜ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="urijština" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A8" title="ਤਿਕੋਨ – paňdžábština" lang="pa" hreflang="pa" data-title="ਤਿਕੋਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="paňdžábština" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Dreieck" title="Dreieck – falčtina" lang="pfl" hreflang="pfl" data-title="Dreieck" data-language-autonym="Pälzisch" data-language-local-name="falčtina" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tr%C3%B3jk%C4%85t" title="Trójkąt – polština" lang="pl" hreflang="pl" data-title="Trójkąt" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%DA%A9%D9%88%D9%86" title="تکون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="تکون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF%D8%B1%DB%90%DA%85%D9%86%DA%89%DB%8C" title="درېڅنډی – paštština" lang="ps" hreflang="ps" data-title="درېڅنډی" data-language-autonym="پښتو" data-language-local-name="paštština" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tri%C3%A2ngulo" title="Triângulo – portugalština" lang="pt" hreflang="pt" data-title="Triângulo" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Kimsak%27uchu" title="Kimsak'uchu – kečuánština" lang="qu" hreflang="qu" data-title="Kimsak'uchu" data-language-autonym="Runa Simi" data-language-local-name="kečuánština" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Triunghi" title="Triunghi – rumunština" lang="ro" hreflang="ro" data-title="Triunghi" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Треугольник – ruština" lang="ru" hreflang="ru" data-title="Треугольник" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%83%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Триуголник – rusínština" lang="rue" hreflang="rue" data-title="Триуголник" data-language-autonym="Русиньскый" data-language-local-name="rusínština" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tri%C3%A0nculu" title="Triànculu – sicilština" lang="scn" hreflang="scn" data-title="Triànculu" data-language-autonym="Sicilianu" data-language-local-name="sicilština" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Triangle" title="Triangle – skotština" lang="sco" hreflang="sco" data-title="Triangle" data-language-autonym="Scots" data-language-local-name="skotština" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%BD%DA%AA%D9%86%DA%8A%D9%88" title="ٽڪنڊو – sindhština" lang="sd" hreflang="sd" data-title="ٽڪنڊو" data-language-autonym="سنڌي" data-language-local-name="sindhština" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Golmma%C4%8Diegat" title="Golmmačiegat – sámština (severní)" lang="se" hreflang="se" data-title="Golmmačiegat" data-language-autonym="Davvisámegiella" data-language-local-name="sámština (severní)" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trokut" title="Trokut – srbochorvatština" lang="sh" hreflang="sh" data-title="Trokut" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%9A%E0%B7%9D%E0%B6%AB" title="ත්‍රිකෝණ – sinhálština" lang="si" hreflang="si" data-title="ත්‍රිකෝණ" data-language-autonym="සිංහල" data-language-local-name="sinhálština" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Triangle" title="Triangle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Triangle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Trojuholn%C3%ADk" title="Trojuholník – slovenština" lang="sk" hreflang="sk" data-title="Trojuholník" data-language-autonym="Slovenčina" data-language-local-name="slovenština" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trikotnik" title="Trikotnik – slovinština" lang="sl" hreflang="sl" data-title="Trikotnik" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kulm%C3%A2h%C3%A2%C5%A1" title="Kulmâhâš – sámština (inarijská)" lang="smn" hreflang="smn" data-title="Kulmâhâš" data-language-autonym="Anarâškielâ" data-language-local-name="sámština (inarijská)" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gonyonhatu" title="Gonyonhatu – šonština" lang="sn" hreflang="sn" data-title="Gonyonhatu" data-language-autonym="ChiShona" data-language-local-name="šonština" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Saddexagal" title="Saddexagal – somálština" lang="so" hreflang="so" data-title="Saddexagal" data-language-autonym="Soomaaliga" data-language-local-name="somálština" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Trek%C3%ABnd%C3%ABshi" title="Trekëndëshi – albánština" lang="sq" hreflang="sq" data-title="Trekëndëshi" data-language-autonym="Shqip" data-language-local-name="albánština" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE" title="Троугао – srbština" lang="sr" hreflang="sr" data-title="Троугао" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Juru_tilu" title="Juru tilu – sundština" lang="su" hreflang="su" data-title="Juru tilu" data-language-autonym="Sunda" data-language-local-name="sundština" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Triangel" title="Triangel – švédština" lang="sv" hreflang="sv" data-title="Triangel" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Pembetatu" title="Pembetatu – svahilština" lang="sw" hreflang="sw" data-title="Pembetatu" data-language-autonym="Kiswahili" data-language-local-name="svahilština" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Trziek" title="Trziek – slezština" lang="szl" hreflang="szl" data-title="Trziek" data-language-autonym="Ślůnski" data-language-local-name="slezština" class="interlanguage-link-target"><span>Ślůnski</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%AE%E0%AF%8D" title="முக்கோணம் – tamilština" lang="ta" hreflang="ta" data-title="முக்கோணம்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%AD%E0%B1%81%E0%B0%9C%E0%B0%82" title="త్రిభుజం – telugština" lang="te" hreflang="te" data-title="త్రిభుజం" data-language-autonym="తెలుగు" data-language-local-name="telugština" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A1%D0%B5%D0%BA%D1%83%D0%BD%D2%B7%D0%B0" title="Секунҷа – tádžičtina" lang="tg" hreflang="tg" data-title="Секунҷа" data-language-autonym="Тоҷикӣ" data-language-local-name="tádžičtina" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%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="รูปสามเหลี่ยม – thajština" lang="th" hreflang="th" data-title="รูปสามเหลี่ยม" data-language-autonym="ไทย" data-language-local-name="thajština" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tatsulok" title="Tatsulok – tagalog" lang="tl" hreflang="tl" data-title="Tatsulok" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gen" title="Üçgen – turečtina" lang="tr" hreflang="tr" data-title="Üçgen" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D3%A8%D1%87%D0%BF%D0%BE%D1%87%D0%BC%D0%B0%D0%BA" title="Өчпочмак – tatarština" lang="tt" hreflang="tt" data-title="Өчпочмак" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarština" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA" title="Трикутник – ukrajinština" lang="uk" hreflang="uk" data-title="Трикутник" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – urdština" lang="ur" hreflang="ur" data-title="مثلث" data-language-autonym="اردو" data-language-local-name="urdština" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Uchburchak" title="Uchburchak – uzbečtina" lang="uz" hreflang="uz" data-title="Uchburchak" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Triango%C5%82o" title="Triangoło – benátština" lang="vec" hreflang="vec" data-title="Triangoło" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Tam_gi%C3%A1c" title="Tam giác – vietnamština" lang="vi" hreflang="vi" data-title="Tam giác" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Drieoek" title="Drieoek – vlámština (západní)" lang="vls" hreflang="vls" data-title="Drieoek" data-language-autonym="West-Vlams" data-language-local-name="vlámština (západní)" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Trayanggulo" title="Trayanggulo – warajština" lang="war" hreflang="war" data-title="Trayanggulo" data-language-autonym="Winaray" data-language-local-name="warajština" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="三角形" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%93%D7%A8%D7%99%D7%99%D7%A2%D7%A7" title="דרייעק – jidiš" lang="yi" hreflang="yi" data-title="דרייעק" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/An%C3%ADgunm%E1%BA%B9%CC%81ta" title="Anígunmẹ́ta – jorubština" lang="yo" hreflang="yo" data-title="Anígunmẹ́ta" data-language-autonym="Yorùbá" data-language-local-name="jorubština" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%8E%E2%B4%BD%E2%B5%95%E2%B4%B0%E2%B4%B9" title="ⴰⵎⴽⵕⴰⴹ – tamazight (standardní marocký)" lang="zgh" hreflang="zgh" data-title="ⴰⵎⴽⵕⴰⴹ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="tamazight (standardní marocký)" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – čínština" lang="zh" hreflang="zh" data-title="三角形" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – čínština (klasická)" lang="lzh" hreflang="lzh" data-title="三角形" data-language-autonym="文言" data-language-local-name="čínština (klasická)" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Sa%E2%81%BF-kak-h%C3%AAng" title="Saⁿ-kak-hêng – čínština (dialekty Minnan)" lang="nan" hreflang="nan" data-title="Saⁿ-kak-hêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="čínština (dialekty Minnan)" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – kantonština" lang="yue" hreflang="yue" data-title="三角形" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</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/Q19821#sitelinks-wikipedia" title="Editovat mezijazykové odkazy" class="wbc-editpage">Upravit odkazy</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="Jmenné prostory"> <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 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id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><div class="uvodni-upozorneni hatnote noprint">Tento článek je o <a href="/wiki/Geometrie" title="Geometrie">geometrii</a>. Další významy jsou uvedeny na stránce <a href="/wiki/Troj%C3%BAheln%C3%ADk_(rozcestn%C3%ADk)" class="mw-disambig" title="Trojúhelník (rozcestník)">Trojúhelník (rozcestník)</a>.</div> <p><b>Trojúhelník</b> (symbol △) je základní <a href="/wiki/Geometrick%C3%BD_%C3%BAtvar" title="Geometrický útvar">geometrický útvar</a>, který má tři <a href="/wiki/Vrchol_(geometrie)" title="Vrchol (geometrie)">vrcholy</a> a tři <a href="/wiki/Strana_(geometrie)" title="Strana (geometrie)">strany</a>. V <a href="/wiki/Eukleidovsk%C3%A1_geometrie" title="Eukleidovská geometrie">euklidovské geometrii</a> jakékoliv tři <a href="/wiki/Bod" title="Bod">body</a> neležící v jedné <a href="/wiki/P%C5%99%C3%ADmka" title="Přímka">přímce</a> určují právě jeden trojúhelník a právě jednu <a href="/wiki/Rovina" title="Rovina">rovinu</a> v dvourozměrném <a href="/wiki/Eukleidovsk%C3%BD_prostor" title="Eukleidovský prostor">euklidovském prostoru</a> (<a href="/wiki/2D" title="2D">2D</a>). Trojúhelník s vrcholy A, B, C označujeme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span>, proti vrcholům leží strany <i>a</i>, <i>b</i>, <i>c</i>, <a href="/wiki/Vnit%C5%99n%C3%AD_%C3%BAhel" class="mw-redirect" title="Vnitřní úhel">vnitřní úhly</a> označujeme řeckými písmeny (<a href="/wiki/Alfa" title="Alfa">α</a>, <a href="/wiki/Beta" title="Beta">β</a>, <a href="/wiki/Gama" title="Gama">γ</a>). Součet vnitřních úhlů takového trojúhelníku je roven 180<a href="/wiki/Stupe%C5%88_(%C3%BAhel)" title="Stupeň (úhel)">°</a> (tj. <a href="/wiki/Radi%C3%A1n" title="Radián">π</a> v obloukové míře). Pokud není uvedeno jinak, týká se článek pouze Euklidovské geometrie a Euklidovské roviny. </p><p>Mimo euklidovskou geometrii existuje <a href="/wiki/Sf%C3%A9rick%C3%BD_troj%C3%BAheln%C3%ADk" class="mw-redirect" title="Sférický trojúhelník">sférický trojúhelník</a> na <a href="/wiki/Sf%C3%A9ra_(matematika)" title="Sféra (matematika)">kulové ploše</a> (např. dostatečně velký trojúhelník vyznačený na povrchu zeměkoule), který má součet velikostí vnitřních úhlů vždy větší než 180° a trojúhelník v <a href="/wiki/Loba%C4%8Devsk%C3%A9ho_geometrie" class="mw-redirect" title="Lobačevského geometrie">hyperbolické (Lobačevského) rovině</a> (tj. trojúhelník na vnitřním povrchu koule) vždy menší než 180°. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Základní_pojmy"><span id="Z.C3.A1kladn.C3.AD_pojmy"></span>Základní pojmy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=1" title="Editace sekce: Základní pojmy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=1" title="Editovat zdrojový kód sekce Základní pojmy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Triangle_-_angles,_vertices,_sides.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Triangle_-_angles%2C_vertices%2C_sides.svg/220px-Triangle_-_angles%2C_vertices%2C_sides.svg.png" decoding="async" width="220" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Triangle_-_angles%2C_vertices%2C_sides.svg/330px-Triangle_-_angles%2C_vertices%2C_sides.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Triangle_-_angles%2C_vertices%2C_sides.svg/440px-Triangle_-_angles%2C_vertices%2C_sides.svg.png 2x" data-file-width="179" data-file-height="160" /></a><figcaption>Trojúhelník <i>ABC</i> s vrcholy <i>A</i>, <i>B</i>, <i>C</i> a stranami <i>a</i>, <i>b</i>, <i>c</i></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Triangle_-_angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle_-_angles.svg/220px-Triangle_-_angles.svg.png" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle_-_angles.svg/330px-Triangle_-_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle_-_angles.svg/440px-Triangle_-_angles.svg.png 2x" data-file-width="243" data-file-height="191" /></a><figcaption>Vnitřní úhly: <i>α, β, γ</i><br />vnější úhly: <i>α’</i>, <i>β’</i>, <i>γ’</i> a <i>α’’</i>, <i>β’’</i>, <i>γ’’</i></figcaption></figure> <p><a href="/wiki/%C3%9Ase%C4%8Dka" title="Úsečka">Úsečky</a>, které spojují <a href="/wiki/Vrchol_(geometrie)" title="Vrchol (geometrie)">vrcholy</a>, se nazývají <a href="/wiki/Strana_(geometrie)" title="Strana (geometrie)">strany</a> trojúhelníku. <a href="/wiki/%C3%9Ahel" title="Úhel">Úhly</a>, které svírají strany, se nazývají <i>vnitřní úhly</i> trojúhelníku. Vedlejší úhly k vnitřním úhlům se nazývají <i>vnější úhly</i> trojúhelníka (u každého vrcholu jsou dva, vnější úhel je doplněk vnitřního úhlu do 180°).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Každý trojúhelník má 3 strany, 3 vnitřní úhly, 6 vnějších úhlů. Trojúhelník nemá <a href="/wiki/%C3%9Ahlop%C5%99%C3%AD%C4%8Dka" title="Úhlopříčka">úhlopříčky</a>. </p><p>Označíme-li vrcholy trojúhelníka velkými tiskacími písmeny A, B, C, pak trojúhelník označujeme jako <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span>. Takový trojúhelník má strany označené malými písmeny <i>a</i>, <i>b</i>, <i>c</i> tak, že stejnojmenná strana leží vždy proti stejně označenému vrcholu (tj. proti vrcholu A leží strana <i>a</i>, proti vrcholu B leží strana <i>b</i> atd.). Vnitřní úhly se označují popořadě <a href="/wiki/%C5%98eck%C3%A9_p%C3%ADsmo" title="Řecké písmo">řeckými písmeny</a> (tj. u vrcholu A leží vnitřní úhel <a href="/wiki/Alfa" title="Alfa">α</a>, u vrcholu B leží <a href="/wiki/Beta" title="Beta">β</a>, u vrcholu C leží <a href="/wiki/Gama" title="Gama">γ</a>). Podobně se postupuje při označení trojúhelníka jinými písmeny, např. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle KLM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>K</mi> <mi>L</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle KLM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2653c02a9bfc0fd87ee00cfc3aa70fa7792fcda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.157ex; height:2.176ex;" alt="{\displaystyle \triangle KLM}"></span>. Při označení trojúhelníka má význam pořadí stran (jedná se obvykle o různé trojúhelníky), což se využívá například u <a href="/wiki/Podobnost_(geometrie)" title="Podobnost (geometrie)">podobnosti trojúhelníků</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Druhy_trojúhelníků"><span id="Druhy_troj.C3.BAheln.C3.ADk.C5.AF"></span>Druhy trojúhelníků</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=2" title="Editace sekce: Druhy trojúhelníků" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=2" title="Editovat zdrojový kód sekce Druhy trojúhelníků"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Eukleid%C3%A9s" title="Eukleidés">Eukleidés</a> napsal již před dvěma tisíci lety knihu <a href="/wiki/Eukleidovy_Z%C3%A1klady" title="Eukleidovy Základy">Základy</a>, kde popisuje trojúhelník jako jeden ze základních geometrických elementů. Druhy trojúhelníků jsou od té doby buď přímými překlady z řečtiny nebo latiny. </p> <div class="mw-heading mw-heading4"><h4 id="Podle_stran">Podle stran</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=3" title="Editace sekce: Podle stran" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=3" title="Editovat zdrojový kód sekce Podle stran"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euklidés definoval podle délky stran tři druhy trojúhelníků: </p> <ul><li>Obecný trojúhelník (též <i>různostranný</i>) – žádné dvě strany nejsou <a href="/wiki/Shodn%C3%A9_zobrazen%C3%AD" title="Shodné zobrazení">shodné</a> (<a href="/wiki/%C5%98e%C4%8Dtina" title="Řečtina">řecky</a> <span class="cizojazycne" lang="el" title="řečtina">σκαληνὸν</span>, čti <i><span title="Toto je přepis výslovnosti pomocí znaků mezinárodní fonetické abecedy (IPA)." class="IPA">skalinón</span></i>, znamená <i>nestejné</i>)</li> <li><a href="/wiki/Rovnoramenn%C3%BD_troj%C3%BAheln%C3%ADk" title="Rovnoramenný trojúhelník">Rovnoramenný trojúhelník</a> – dvě strany jsou navzájem shodné, ale nejsou shodné s třetí stranou (<a href="/wiki/%C5%98e%C4%8Dtina" title="Řečtina">řecky</a> <span class="cizojazycne" lang="el" title="řečtina">ἰσοσκελὲς</span>, čti <i><span title="Toto je přepis výslovnosti pomocí znaků mezinárodní fonetické abecedy (IPA)." class="IPA">isoskelés</span></i>, znamená <i>stejné nohy</i>)</li> <li><a href="/wiki/Rovnostrann%C3%BD_troj%C3%BAheln%C3%ADk" title="Rovnostranný trojúhelník">Rovnostranný trojúhelník</a> – všechny strany jsou shodné (<a href="/wiki/%C5%98e%C4%8Dtina" title="Řečtina">řecky</a> <span class="cizojazycne" lang="el" title="řečtina">ἰσόπλευρον</span>, čti <i><span title="Toto je přepis výslovnosti pomocí znaků mezinárodní fonetické abecedy (IPA)." class="IPA">isópleuron</span></i>, znamená <i>stejné strany</i>)</li></ul> <table> <tbody><tr style="text-align:bottom"> <td><span typeof="mw:File"><a href="/wiki/Soubor:Triangle.Scalene.svg" class="mw-file-description" title="Obecný trojúhelník"><img alt="Obecný trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/245px-Triangle.Scalene.svg.png" decoding="async" width="245" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/368px-Triangle.Scalene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/490px-Triangle.Scalene.svg.png 2x" data-file-width="245" data-file-height="110" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Soubor:Triangle.Equilateral.svg" class="mw-file-description" title="Rovnostranný trojúhelník"><img alt="Rovnostranný trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/100px-Triangle.Equilateral.svg.png" decoding="async" width="100" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/150px-Triangle.Equilateral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/200px-Triangle.Equilateral.svg.png 2x" data-file-width="512" data-file-height="415" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Soubor:Triangle.Isosceles.svg" class="mw-file-description" title="Rovnoramenný trojúhelník"><img alt="Rovnoramenný trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/75px-Triangle.Isosceles.svg.png" decoding="async" width="75" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/113px-Triangle.Isosceles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/150px-Triangle.Isosceles.svg.png 2x" data-file-width="74" data-file-height="114" /></a></span> </td></tr> <tr align="center"> <th>Obecný </th> <th>Rovnostranný </th> <th>Rovnoramenný </th></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Podle_úhlů"><span id="Podle_.C3.BAhl.C5.AF"></span>Podle úhlů</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=4" title="Editace sekce: Podle úhlů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=4" title="Editovat zdrojový kód sekce Podle úhlů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ostro%C3%BAhl%C3%BD_troj%C3%BAheln%C3%ADk" title="Ostroúhlý trojúhelník">Ostroúhlý trojúhelník</a> – všechny vnitřní úhly jsou ostré</li> <li><a href="/wiki/Pravo%C3%BAhl%C3%BD_troj%C3%BAheln%C3%ADk" title="Pravoúhlý trojúhelník">Pravoúhlý trojúhelník</a> – jeden vnitřní úhel je pravý, zbývající dva jsou ostré</li> <li><a href="/wiki/Tupo%C3%BAhl%C3%BD_troj%C3%BAheln%C3%ADk" title="Tupoúhlý trojúhelník">Tupoúhlý trojúhelník</a> – jeden vnitřní úhel je tupý, zbývající dva jsou ostré</li></ul> <table> <tbody><tr style="text-align:bottom"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Soubor:Triangle.Acute.svg" class="mw-file-description" title="Ostroúhlý trojúhelník"><img alt="Ostroúhlý trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/794px-Triangle.Acute.svg.png" decoding="async" width="794" height="491" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/1191px-Triangle.Acute.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/1588px-Triangle.Acute.svg.png 2x" data-file-width="794" data-file-height="491" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Soubor:Triangle.Right.svg" class="mw-file-description" title="Pravoúhlý trojúhelník"><img alt="Pravoúhlý trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/150px-Triangle.Right.svg.png" decoding="async" width="150" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/225px-Triangle.Right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/300px-Triangle.Right.svg.png 2x" data-file-width="150" data-file-height="113" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Soubor:Triangle.Obtuse.svg" class="mw-file-description" title="Tupoúhlý trojúhelník"><img alt="Tupoúhlý trojúhelník" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/113px-Triangle.Obtuse.svg.png" decoding="async" width="113" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/170px-Triangle.Obtuse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/226px-Triangle.Obtuse.svg.png 2x" data-file-width="113" data-file-height="113" /></a></span> </td></tr> <tr align="center"> <th>Ostroúhlý </th> <th>Pravoúhlý </th> <th>Tupoúhlý </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Obvod_a_obsah">Obvod a obsah</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=5" title="Editace sekce: Obvod a obsah" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=5" title="Editovat zdrojový kód sekce Obvod a obsah"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Obvod trojúhelníku <i>o</i> vypočteme jako součet všech jeho stran: </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 o=a+b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o=a+b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e00029e4959eab4cc3c9ea8a56974caf3432b6a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.141ex; height:2.343ex;" alt="{\displaystyle o=a+b+c}"></span> , kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> jsou strany trojúhelníku</dd></dl> <p><a href="/wiki/Obsah" title="Obsah">Obsah</a> trojúhelníku <i>S</i> se vypočte jako polovina součinu libovolné strany a k ní příslušné výšky: </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 S=v_{a}\cdot {a \over 2}={v_{a}\cdot a \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=v_{a}\cdot {a \over 2}={v_{a}\cdot a \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d633169bb3bdcd39777b8f87da8c12a8ea7f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.645ex; height:4.676ex;" alt="{\displaystyle S=v_{a}\cdot {a \over 2}={v_{a}\cdot a \over 2}}"></span>, kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca1db88e93760becec7c732d1431d7b2a56f8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.229ex; height:2.009ex;" alt="{\displaystyle v_{a}}"></span> je výška příslušná straně <i>a</i></dd></dl> <p>Pokud není známá příslušná výška, je možné obsah trojúhelníku vypočítat podle <a href="/wiki/Heron%C5%AFv_vzorec" title="Heronův vzorec">Heronova vzorce</a> </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 s={o \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>o</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={o \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e155c5fb20db54c6428da41a69b5e2576fe16506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.188ex; height:4.676ex;" alt="{\displaystyle s={o \over 2}}"></span>, kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> je obvod trojúhelníku</dd></dl> <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 S={\sqrt {s(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6596e9344ada420302022229b1d8122cc7a3141c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.467ex; height:4.843ex;" alt="{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}"></span></dd></dl> <p>Obsah trojúhelníku pomocí poloměru kružnice opsané (<span class="mwe-math-element"><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/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></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 S={\frac {abc}{4r}}=2r^{2}\sin \alpha \sin \beta \sin \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> <mrow> <mn>4</mn> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {abc}{4r}}=2r^{2}\sin \alpha \sin \beta \sin \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be4054ae603e757e4245ba7e585467f93264228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.003ex; height:5.343ex;" alt="{\displaystyle S={\frac {abc}{4r}}=2r^{2}\sin \alpha \sin \beta \sin \gamma }"></span></dd></dl> <p>Obsah trojúhelníku pomocí poloměru kružnice vepsané (<span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></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 S={\frac {a+b+c}{2}}\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {a+b+c}{2}}\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc684c1f641a63fc782d34293720bed784a11168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.551ex; height:5.343ex;" alt="{\displaystyle S={\frac {a+b+c}{2}}\rho }"></span></dd></dl> <p>Obsah trojúhelníku pomocí vnitřního úhlu: </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 S={\frac {1}{2}}ab\,\sin \gamma ={\frac {1}{2}}ac\,\sin \beta ={\frac {1}{2}}bc\,\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>c</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>b</mi> <mi>c</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}ab\,\sin \gamma ={\frac {1}{2}}ac\,\sin \beta ={\frac {1}{2}}bc\,\sin \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f13c23c1097f2f63561ae69567c613c18fed5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.391ex; height:5.176ex;" alt="{\displaystyle S={\frac {1}{2}}ab\,\sin \gamma ={\frac {1}{2}}ac\,\sin \beta ={\frac {1}{2}}bc\,\sin \alpha }"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Troj%C3%BAheln%C3%ADk_ABC-D.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Troj%C3%BAheln%C3%ADk_ABC-D.jpg/220px-Troj%C3%BAheln%C3%ADk_ABC-D.jpg" decoding="async" width="220" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Troj%C3%BAheln%C3%ADk_ABC-D.jpg/330px-Troj%C3%BAheln%C3%ADk_ABC-D.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Troj%C3%BAheln%C3%ADk_ABC-D.jpg/440px-Troj%C3%BAheln%C3%ADk_ABC-D.jpg 2x" data-file-width="518" data-file-height="765" /></a><figcaption>Trojúhelník_ABC-D</figcaption></figure> <p>Obsah obecného trojúhelníku v rovině kde (a<sub>x</sub>,a<sub>y</sub>), (b<sub>x</sub>,b<sub>y</sub>), (c<sub>x</sub>,c<sub>y</sub>) jsou souřadnice vrcholů (vychází z vektorového součinu, použití hlavně v grafice): </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 S={\frac {1}{2}}|(c_{x}-a_{x})(b_{y}-a_{y})-(c_{y}-a_{y})(b_{x}-a_{x})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}|(c_{x}-a_{x})(b_{y}-a_{y})-(c_{y}-a_{y})(b_{x}-a_{x})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcac0385cd194aa775d9108db3b08271d6a9b79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.144ex; height:5.176ex;" alt="{\displaystyle S={\frac {1}{2}}|(c_{x}-a_{x})(b_{y}-a_{y})-(c_{y}-a_{y})(b_{x}-a_{x})|}"></span></dd></dl> <p>nebo: </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 S={\frac {1}{2}}|[a_{x}(c_{y}-b_{y})+b_{x}(a_{y}-c_{y})+c_{x}(b_{y}-a_{y})]|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}|[a_{x}(c_{y}-b_{y})+b_{x}(a_{y}-c_{y})+c_{x}(b_{y}-a_{y})]|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295f71a4382867199d71951f0ff196e85ba42b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.329ex; height:5.176ex;" alt="{\displaystyle S={\frac {1}{2}}|[a_{x}(c_{y}-b_{y})+b_{x}(a_{y}-c_{y})+c_{x}(b_{y}-a_{y})]|}"></span></dd></dl> <p>Použije-li se předcházející vzorec bez absolutní hodnoty, lze jej využít pro ověření zda bod (d<sub>x</sub>,d<sub>y</sub>) leží uvnitř trojúhelníku ABC. V případě, že leží, tak <a href="/wiki/Funkce_signum" title="Funkce signum">znaménka</a> obsahů všech čtyř trojúhelníků ABC, ABD, BCD a CAD jsou stejná. Leží-li vně, nemají všechny obsahy stejné znaménko. To je kladné, obíhájí-li vrcholy ve směru hodinových ručiček. </p> <div class="mw-heading mw-heading2"><h2 id="Věty_o_trojúhelníku"><span id="V.C4.9Bty_o_troj.C3.BAheln.C3.ADku"></span>Věty o trojúhelníku</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=6" title="Editace sekce: Věty o trojúhelníku" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=6" title="Editovat zdrojový kód sekce Věty o trojúhelníku"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V <a href="/wiki/Pravo%C3%BAhl%C3%BD_troj%C3%BAheln%C3%ADk" title="Pravoúhlý trojúhelník">pravoúhlém trojúhelníku</a> platí: </p> <ul><li><a href="/wiki/Pythagorova_v%C4%9Bta" title="Pythagorova věta">Pythagorova věta</a> (c² = a² + b²)</li></ul> <ul><li><a href="/wiki/Eukleidova_v%C4%9Bta_o_v%C3%BD%C5%A1ce" title="Eukleidova věta o výšce">Euklidova věta o výšce</a></li></ul> <p>V jakémkoliv obecném trojúhelníku platí: </p> <ul><li><a href="/wiki/Sinov%C3%A1_v%C4%9Bta" title="Sinová věta">sinová věta</a></li> <li><a href="/wiki/Kosinov%C3%A1_v%C4%9Bta" title="Kosinová věta">kosinová věta</a> (zobecnění <a href="/wiki/Pythagorova_v%C4%9Bta" title="Pythagorova věta">Pythagorovy věty</a> na nepravoúhlý trojúhelník)</li> <li><a href="/wiki/Tangentov%C3%A1_v%C4%9Bta" title="Tangentová věta">tangentová věta</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti_trojúhelníku"><span id="Vlastnosti_troj.C3.BAheln.C3.ADku"></span>Vlastnosti trojúhelníku</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=7" title="Editace sekce: Vlastnosti trojúhelníku" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=7" title="Editovat zdrojový kód sekce Vlastnosti trojúhelníku"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Trojúhelníková_nerovnost"><span id="Troj.C3.BAheln.C3.ADkov.C3.A1_nerovnost"></span>Trojúhelníková nerovnost</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=8" title="Editace sekce: Trojúhelníková nerovnost" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=8" title="Editovat zdrojový kód sekce Trojúhelníková nerovnost"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Strana_(geometrie)" title="Strana (geometrie)">Strany</a></i> trojúhelníku splňují větu o <a href="/wiki/Troj%C3%BAheln%C3%ADkov%C3%A1_nerovnost" title="Trojúhelníková nerovnost">trojúhelníkové nerovnosti</a>. Součet dvou libovolných stran trojúhelníku je vždy <i>delší</i> než strana třetí, neboli: </p> <dl><dd><i>a</i> + <i>b</i> > <i>c</i></dd> <dd><i>a</i> + <i>c</i> > <i>b</i></dd> <dd><i>b</i> + <i>c</i> > a</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a-b|<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-b|<c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88da9db8076ac01f03f88100a5953d815f9a1fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.467ex; height:2.843ex;" alt="{\displaystyle |a-b|<c}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a-c|<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-c|<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52c8635b7ed3a4e6d75a9952b0f1923a3828222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.467ex; height:2.843ex;" alt="{\displaystyle |a-c|<b}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b-c|<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b-c|<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82251b390f6c8384ed179fc9be10c3864d8a92a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.467ex; height:2.843ex;" alt="{\displaystyle |b-c|<a}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Součet_úhlů"><span id="Sou.C4.8Det_.C3.BAhl.C5.AF"></span>Součet úhlů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=9" title="Editace sekce: Součet úhlů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=9" title="Editovat zdrojový kód sekce Součet úhlů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Součet všech <i>vnitřních úhlů</i> je v každém trojúhelníku 180°. Součet vnitřního a příslušného vnějšího úhlu je 180°. Součet dvou vnitřních úhlů se rovná vnějšímu úhlu u zbývajícího vrcholu. Proti většímu úhlu leží větší strana. </p> <dl><dd><i>α</i> + <i>α’</i> = <i>β</i> + <i>β’</i> = <i>γ</i> + <i>γ’</i> = 180°</dd></dl> <dl><dd><i>α</i> + <i>β</i> = <i>γ’</i></dd> <dd><i>α</i> + <i>γ</i> = <i>β’</i></dd> <dd><i>β</i> + <i>γ</i>= <i>α’</i></dd></dl> <dl><dd><i>α</i> + <i>β</i> + <i>γ</i> = 180°</dd> <dd><i>α’</i> + <i>β’</i> + <i>γ’</i> = 360°</dd></dl> <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 \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78731cad789eccabfe272fc5d2a85dbd1c044da6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.998ex; height:5.343ex;" alt="{\displaystyle \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \alpha +\cos \beta +\cos \gamma =1+4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \alpha +\cos \beta +\cos \gamma =1+4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1972917be48eed4f3acd367334ead1f1703787ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.001ex; height:5.343ex;" alt="{\displaystyle \cos \alpha +\cos \beta +\cos \gamma =1+4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2(1+\cos \alpha \cos \beta \cos \gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2(1+\cos \alpha \cos \beta \cos \gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a987870bf6175927d144d09710481007f312bed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.077ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2(1+\cos \alpha \cos \beta \cos \gamma )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =1-2\cos \alpha \cos \beta \cos \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =1-2\cos \alpha \cos \beta \cos \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6334ed72ab4ee3028121438921970c2a7eea8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.422ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =1-2\cos \alpha \cos \beta \cos \gamma }"></span></dd></dl> <p>Zavedeme-li veličinu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {1}{2}}(a+b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {1}{2}}(a+b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd057f5a01447ab6dfeb670f5f2a410131ba71a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.912ex; height:5.176ex;" alt="{\displaystyle s={\frac {1}{2}}(a+b+c)}"></span>, pak lze velikosti vnitřních úhlů určit ze vztahů </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 \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3daf3f1a467f76953096397b6577fdf8d03e931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.31ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {(s-c)(s-a)}{ac}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {(s-c)(s-a)}{ac}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79118cc143ee821a7f96fa0c6b9073e73bb1668e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.386ex; height:7.676ex;" alt="{\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {(s-c)(s-a)}{ac}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {(s-a)(s-b)}{ab}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {(s-a)(s-b)}{ab}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb86a2a184c3762bb3cc85e80bc56314b5ddb19f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.307ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {(s-a)(s-b)}{ab}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8ecf93f3cdfab4f4a1ea7d4d2536413ef578cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.141ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s(s-b)}{ac}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s(s-b)}{ac}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2511b2a8ffcad1fa704ebe267cda59ee0c3191a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.753ex; height:7.676ex;" alt="{\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s(s-b)}{ac}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s(s-c)}{ab}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s(s-c)}{ab}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47f73cb85162fed8ddbf90737789a2d3dd8f70e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.692ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s(s-c)}{ab}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Souměrnost"><span id="Soum.C4.9Brnost"></span>Souměrnost</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=10" title="Editace sekce: Souměrnost" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=10" title="Editovat zdrojový kód sekce Souměrnost"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Obecný trojúhelník není <a href="/wiki/Osov%C3%A1_soum%C4%9Brnost" title="Osová souměrnost">osově</a> ani <a href="/wiki/St%C5%99edov%C3%A1_soum%C4%9Brnost" title="Středová souměrnost">středově souměrný</a>, některé druhy trojúhelníků mohou být <a href="/wiki/Osov%C3%A1_soum%C4%9Brnost" title="Osová souměrnost">osově souměrné</a> např.: rovnoramenný a rovnostranný trojúhelník. </p> <div class="mw-heading mw-heading3"><h3 id="Výška"><span id="V.C3.BD.C5.A1ka"></span>Výška</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=11" title="Editace sekce: Výška" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=11" title="Editovat zdrojový kód sekce Výška"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Soubor:VyskyTrojuhelnika.jpg" class="mw-file-description" title="Výška trojúhelníku"><img alt="Výška trojúhelníku" src="//upload.wikimedia.org/wikipedia/commons/d/d2/VyskyTrojuhelnika.jpg" decoding="async" width="509" height="203" class="mw-file-element" data-file-width="509" data-file-height="203" /></a><figcaption>Výška trojúhelníku</figcaption></figure> <p>Výška je úsečka, jejímiž krajními body jsou vrchol trojúhelníku a pata kolmice vedené tímto vrcholem na protější stranu, resp. přímku na níž protější strana leží (nebo, stručněji řečeno, kolmice z vrcholu na protější stranu, resp. přímku na níž protější strana leží). Průsečík výšky s příslušnou stranou se nazývá <i>pata výšky</i>. Každý trojúhelník má tři výšky. </p><p>Přímky, na nichž leží výšky, se protínají v jednom bodě, který se nazývá <i>ortocentrum</i>. Ortocentrum leží buď uvnitř trojúhelníku, pokud je ostroúhlý, nebo u pravoúhlého trojúhelníka splývá s jeho vrcholem, při němž je pravý úhel, anebo leží vně, je-li to trojúhelník tupoúhlý. </p><p>Spojnice jednotlivých pat výšek tvoří <a href="/wiki/Ortick%C3%BD_troj%C3%BAheln%C3%ADk" title="Ortický trojúhelník">ortický trojúhelník</a>. Pravoúhlý trojúhelník svůj ortický trojúhelník nemá, protože jeho dvě paty výšek splývají. </p><p>Ortocentrum tupoúhlého trojúhelníku je středem jedné z <a href="/wiki/Kru%C5%BEnice_p%C5%99ipsan%C3%A1" title="Kružnice připsaná">kružnic připsaných</a> jeho ortickému trojúhelníku. </p><p>Výšky se označují malým písmenem <i>v</i> s dolním indexem příslušné strany. </p><p>Pro výšky trojúhelníku platí: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{a}:v_{b}:v_{c}={\frac {1}{a}}:{\frac {1}{b}}:{\frac {1}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{a}:v_{b}:v_{c}={\frac {1}{a}}:{\frac {1}{b}}:{\frac {1}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9e4390bed86a9352f91b850edc67d9c51b60e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.277ex; height:5.343ex;" alt="{\displaystyle v_{a}:v_{b}:v_{c}={\frac {1}{a}}:{\frac {1}{b}}:{\frac {1}{c}}}"></span>, čili: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{a}.a=v_{b}.b=v_{c}.c=2S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> <mi>a</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>.</mo> <mi>b</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>.</mo> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{a}.a=v_{b}.b=v_{c}.c=2S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a506e5ace648d8a108217f59b75c39cda596121c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.66ex; height:2.509ex;" alt="{\displaystyle v_{a}.a=v_{b}.b=v_{c}.c=2S}"></span>, kde S je plocha trojúhelníka. </p><p>Velikosti výšek jsou určeny vztahy </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 v_{a}=b\sin \gamma =c\sin \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{a}=b\sin \gamma =c\sin \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c0fe90b858771f69ec403ffeaa0478a035c5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.285ex; height:2.676ex;" alt="{\displaystyle v_{a}=b\sin \gamma =c\sin \beta }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{b}=a\sin \gamma =c\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{b}=a\sin \gamma =c\sin \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d4f375d2e65d1f067950a04704ad3a46bbde57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.508ex; height:2.676ex;" alt="{\displaystyle v_{b}=a\sin \gamma =c\sin \alpha }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{c}=a\sin \beta =b\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>=</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{c}=a\sin \beta =b\sin \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e166465642669b94d5a0583b5c27a879da7683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.575ex; height:2.509ex;" alt="{\displaystyle v_{c}=a\sin \beta =b\sin \alpha }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Těžnice"><span id="T.C4.9B.C5.BEnice"></span>Těžnice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=12" title="Editace sekce: Těžnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=12" title="Editovat zdrojový kód sekce Těžnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Soubor:TezniceTrojuhelnika.jpg" class="mw-file-description" title="Těžnice trojúhelníku"><img alt="Těžnice trojúhelníku" src="//upload.wikimedia.org/wikipedia/commons/5/5a/TezniceTrojuhelnika.jpg" decoding="async" width="498" height="205" class="mw-file-element" data-file-width="498" data-file-height="205" /></a><figcaption>Těžnice trojúhelníku</figcaption></figure> <p><a href="/wiki/T%C4%9B%C5%BEnice" title="Těžnice">Těžnice</a> je úsečka, jejímiž krajními body jsou střed strany a protilehlý vrchol trojúhelníku. Každý trojúhelník má tři těžnice. Těžnice se protínají v jednom bodě, který se nazývá <a href="/wiki/T%C4%9B%C5%BEi%C5%A1t%C4%9B" title="Těžiště">těžiště</a>. Těžiště rozděluje každou těžnici na dva díly v poměru 2 : 1, přitom vzdálenost těžiště od vrcholu je dvojnásobek vzdálenosti od středu protější strany. Každá těžnice rozděluje trojúhelník na dva díly se <i>stejným obsahem</i>. Těžnice se označují malým písmenem <i>t</i> s dolním indexem příslušné strany, těžiště se označuje písmenem <i>T</i>. Těžiště a dva vrcholy trojúhelníku tvoří postupně tři trojúhelníky (ABT, ACT, CBT), všechny tři mají <i>stejný obsah</i>. </p><p>Souřadnice těžiště lze vypočítat ze souřadnic bodů <i>A</i>,<i>B</i>,<i>C</i> podle vztahu: </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 T={\frac {A+B+C}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {A+B+C}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb36054e49a0572fadf88d5c544bd6c8d4999f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.172ex; height:5.343ex;" alt="{\displaystyle T={\frac {A+B+C}{3}}.}"></span></dd></dl> <p>Délky těžnic jsou </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 t_{a}={\frac {1}{2}}{\sqrt {2(b^{2}+c^{2})-a^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}={\frac {1}{2}}{\sqrt {2(b^{2}+c^{2})-a^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e7dc757f8e80675c6bac859da640866a568993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.059ex; height:5.176ex;" alt="{\displaystyle t_{a}={\frac {1}{2}}{\sqrt {2(b^{2}+c^{2})-a^{2}}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{b}={\frac {1}{2}}{\sqrt {2(a^{2}+c^{2})-b^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{b}={\frac {1}{2}}{\sqrt {2(a^{2}+c^{2})-b^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292690543350940036bdef47dc7edb3bc38c9255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.894ex; height:5.176ex;" alt="{\displaystyle t_{b}={\frac {1}{2}}{\sqrt {2(a^{2}+c^{2})-b^{2}}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{c}={\frac {1}{2}}{\sqrt {2(a^{2}+b^{2})-c^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{c}={\frac {1}{2}}{\sqrt {2(a^{2}+b^{2})-c^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df45fb5d91b80434fa0b9c10182e57f497b9da77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.901ex; height:5.176ex;" alt="{\displaystyle t_{c}={\frac {1}{2}}{\sqrt {2(a^{2}+b^{2})-c^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Střední_příčka"><span id="St.C5.99edn.C3.AD_p.C5.99.C3.AD.C4.8Dka"></span>Střední příčka</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=13" title="Editace sekce: Střední příčka" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=13" title="Editovat zdrojový kód sekce Střední příčka"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Soubor:StredniPrickyTrojuhelnika.jpg" class="mw-file-description" title="Střední příčka trojúhelníku"><img alt="Střední příčka trojúhelníku" src="//upload.wikimedia.org/wikipedia/commons/c/c6/StredniPrickyTrojuhelnika.jpg" decoding="async" width="520" height="211" class="mw-file-element" data-file-width="520" data-file-height="211" /></a><figcaption>Střední příčka trojúhelníku</figcaption></figure> <p><i>Střední příčka</i> je spojnice středů dvou stran (dvou <i>pat těžnic</i>). Každý trojúhelník má tři střední příčky. Střední příčka je rovnoběžná s příslušnou stranou a má velikost poloviny příslušné strany. Střední příčky dohromady rozdělují trojúhelník na čtyři shodné trojúhelníky – <i>příčkový trojúhelník</i> a tři trojúhelníky při jednotlivých vrcholech. Těžiště trojúhelníku je zároveň těžištěm jeho příčkového trojúhelníku. Střední příčky se označují malým písmenem <i>s</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Symediána"><span id="Symedi.C3.A1na"></span>Symediána</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=14" title="Editace sekce: Symediána" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=14" title="Editovat zdrojový kód sekce Symediána"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Symediána je osově souměrný obraz těžnice podle osy příslušného úhlu (např. symediána těžnice z vrcholu <i>A</i> podle osy úhlu při vrcholu <i>A</i>). Každý trojúhelník má tři symediány. Všechny symediány trojúhelníku se protínají v jednom bodě, který se nazývá <a href="/wiki/Lemoin%C5%AFv_bod" class="mw-redirect" title="Lemoinův bod">Lemoinův bod</a>. Lemoinův bod leží uvnitř trojúhelníku a platí pro něj, že má ze všech vnitřních bodů trojúhelníku nejmenší součet čtverců vzdáleností od stran trojúhelníku. Pokud Lemoinovým bodem vedeme rovnoběžky s jednotlivými stranami, všechny průsečíky těchto rovnoběžek se stranami (je jich šest) leží na kružnici, která se nazývá <a href="/wiki/Lemoinova_kru%C5%BEnice" title="Lemoinova kružnice">první Lemoinova kružnice</a>. Střed první Lemoinovy kružnice je středem úsečky spojující Lemoinův bod a střed kružnice opsané. </p> <div class="mw-heading mw-heading3"><h3 id="Osy_úhlů"><span id="Osy_.C3.BAhl.C5.AF"></span>Osy úhlů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=15" title="Editace sekce: Osy úhlů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=15" title="Editovat zdrojový kód sekce Osy úhlů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Trojuhelnik_osy.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trojuhelnik_osy.svg/220px-Trojuhelnik_osy.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trojuhelnik_osy.svg/330px-Trojuhelnik_osy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trojuhelnik_osy.svg/440px-Trojuhelnik_osy.svg.png 2x" data-file-width="521" data-file-height="482" /></a><figcaption>Výška, osa úhlu a těžnice v trojúhelníku.</figcaption></figure> <p><i>Osa vnitřního úhlu</i> dělí na polovinu vnitřní úhel a současně protější stranu dělí v poměru délek přilehlých stran. Osa vnějšího úhlu dělí na polovinu vnější úhel. Na <a href="/wiki/Soubor:Trojuhelnik_osy.svg" title="Soubor:Trojuhelnik osy.svg">obrázku</a> je osa vnitřního úhlu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e86ce5904ddc19ee1dbef0834166fab56e9024f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.072ex; height:2.009ex;" alt="{\displaystyle o_{c}}"></span>, osa vnějšího úhlu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o_{c}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o_{c}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60b311edd8c0ea37a39210d2a855fbc9b629f12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.072ex; height:2.509ex;" alt="{\displaystyle o_{c}^{\prime }}"></span> a také těžnice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c989caa6ba66c3afb99ebf759b9755a9d35c980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.784ex; height:2.343ex;" alt="{\displaystyle t_{c}}"></span> a výška <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780b816790d75e7641e3e8949591231e76dc5624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.072ex; height:2.009ex;" alt="{\displaystyle v_{c}}"></span> z vrcholu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. Podobně lze získat osy i u ostatních vrcholů. </p> <div class="mw-heading mw-heading3"><h3 id="Třetiny_úhlů"><span id="T.C5.99etiny_.C3.BAhl.C5.AF"></span>Třetiny úhlů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=16" title="Editace sekce: Třetiny úhlů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=16" title="Editovat zdrojový kód sekce Třetiny úhlů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Morley_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/220px-Morley_triangle.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/330px-Morley_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/440px-Morley_triangle.svg.png 2x" data-file-width="450" data-file-height="300" /></a><figcaption><a href="/w/index.php?title=Morley%C5%AFv_troj%C3%BAheln%C3%ADk&action=edit&redlink=1" class="new" title="Morleyův trojúhelník (stránka neexistuje)">Morleyův trojúhelník</a> je rovnostranný. Vytváří ho přímky dělící vnitřní úhly na třetiny.</figcaption></figure> <p>Pokud jednotlivé vnitřní úhly rozdělíme přímkami na tři stejné díly (<a href="/wiki/Trisekce_%C3%BAhlu" title="Trisekce úhlu">trisekce úhlu</a>), průsečíky těchto přímek (vždy těch dvou, které jsou bližší dané straně trojúhelníku) vždy tvoří rovnostranný trojúhelník (<i>Morleyova věta</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Osy_stran">Osy stran</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=17" title="Editace sekce: Osy stran" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=17" title="Editovat zdrojový kód sekce Osy stran"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Osa strany</i> je kolmice vedená ze středu strany. Osy stran se protínají v jednom bodě (tento bod má stejnou vzdálenost od všech tří vrcholů trojúhelníka). </p> <div class="mw-heading mw-heading3"><h3 id="Eulerova_přímka"><span id="Eulerova_p.C5.99.C3.ADmka"></span>Eulerova přímka</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=18" title="Editace sekce: Eulerova přímka" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=18" title="Editovat zdrojový kód sekce Eulerova přímka"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Eulerova_p%C5%99%C3%ADmka" title="Eulerova přímka">Eulerova přímka</a> je přímka, která prochází těžištěm a ortocentrem a středem <a href="/wiki/Kru%C5%BEnice_opsan%C3%A1" title="Kružnice opsaná">opsané kružnice</a>. Na Eulerově přímce leží i střed <a href="/wiki/Kru%C5%BEnice_opsan%C3%A1" title="Kružnice opsaná">kružnice opsané</a> a střed <a href="/wiki/Kru%C5%BEnice_dev%C3%ADti_bod%C5%AF" title="Kružnice devíti bodů">kružnice devíti bodů</a>. V rovnostranném trojúhelníku těžiště a ortocentrum splývají, takový trojúhelník Eulerovu přímku nemá. </p> <div class="mw-heading mw-heading3"><h3 id="Gaussova_přímka"><span id="Gaussova_p.C5.99.C3.ADmka"></span>Gaussova přímka</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=19" title="Editace sekce: Gaussova přímka" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=19" title="Editovat zdrojový kód sekce Gaussova přímka"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Gauss_line_with_description.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Gauss_line_with_description.svg/220px-Gauss_line_with_description.svg.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Gauss_line_with_description.svg/330px-Gauss_line_with_description.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Gauss_line_with_description.svg/440px-Gauss_line_with_description.svg.png 2x" data-file-width="215" data-file-height="163" /></a><figcaption>Gaussova přímka <i>g</i> se odvozuje popsaným způsobem od trojúhelníku <i>ABC</i> a přímky <i>p.</i></figcaption></figure> <p>Pokud přímka <i>p</i> protíná přímky, na nichž leží strany obecného trojúhelníku, v bodech <i>X, Y, Z,</i> pak středy úseček <i>AX, BY, CZ</i> leží na přímce. Tato přímka <i>g</i> se nazývá <i>Gaussova přímka</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Kružnice_opsaná"><span id="Kru.C5.BEnice_opsan.C3.A1"></span>Kružnice opsaná</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=20" title="Editace sekce: Kružnice opsaná" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=20" title="Editovat zdrojový kód sekce Kružnice opsaná"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:KruzniceOpsanaTrojuhelniku.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/KruzniceOpsanaTrojuhelniku.jpg/220px-KruzniceOpsanaTrojuhelniku.jpg" decoding="async" width="220" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/KruzniceOpsanaTrojuhelniku.jpg/330px-KruzniceOpsanaTrojuhelniku.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/KruzniceOpsanaTrojuhelniku.jpg/440px-KruzniceOpsanaTrojuhelniku.jpg 2x" data-file-width="604" data-file-height="286" /></a><figcaption>Opsaná kružnice</figcaption></figure> <p><a href="/wiki/Kru%C5%BEnice_opsan%C3%A1" title="Kružnice opsaná">Kružnice opsaná</a> trojúhelníku je <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a>, která prochází všemi vrcholy trojúhelníku. Každému trojúhelníku lze opsat kružnici. <i>Střed</i> kružnice opsané leží v <i>průsečíku os stran</i>, <a href="/wiki/Polom%C4%9Br" title="Poloměr">poloměr</a> se rovná vzdálenosti středu od libovolného vrcholu. Spojnice středu kružnice opsané a jednotlivých vrcholů trojúhelníku jsou kolmé k jednotlivým stranám jeho ortického trojúhelníku (tzv. <i>Nagelova věta</i>). </p><p>Velikost <a href="/wiki/Polom%C4%9Br" title="Poloměr">poloměru</a> opsané <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a> určuje vztah </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 r={\frac {a}{2\sin \alpha }}={\frac {b}{2\sin \beta }}={\frac {c}{2\sin \gamma }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {a}{2\sin \alpha }}={\frac {b}{2\sin \beta }}={\frac {c}{2\sin \gamma }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ce3365d675b88b94d515f1bf71061ce1ca7669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.958ex; height:5.843ex;" alt="{\displaystyle r={\frac {a}{2\sin \alpha }}={\frac {b}{2\sin \beta }}={\frac {c}{2\sin \gamma }}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kružnice_vepsaná"><span id="Kru.C5.BEnice_vepsan.C3.A1"></span>Kružnice vepsaná</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=21" title="Editace sekce: Kružnice vepsaná" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=21" title="Editovat zdrojový kód sekce Kružnice vepsaná"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:KruzniceVepsanaTrojuhelniku.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/KruzniceVepsanaTrojuhelniku.jpg/220px-KruzniceVepsanaTrojuhelniku.jpg" decoding="async" width="220" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/KruzniceVepsanaTrojuhelniku.jpg/330px-KruzniceVepsanaTrojuhelniku.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/KruzniceVepsanaTrojuhelniku.jpg/440px-KruzniceVepsanaTrojuhelniku.jpg 2x" data-file-width="615" data-file-height="217" /></a><figcaption>Vepsaná kružnice</figcaption></figure> <p><a href="/wiki/Kru%C5%BEnice_vepsan%C3%A1" title="Kružnice vepsaná">Kružnice vepsaná</a> trojúhelníku je kružnice, která se dotýká všech stran trojúhelníku. Každému trojúhelníku lze vepsat kružnici. Střed kružnice vepsané leží v průsečíku <a href="/wiki/%C3%9Ahel#Souměrnost" title="Úhel">os vnitřních úhlů</a>, <i>poloměr</i> se rovná kolmé vzdálenosti středu od libovolné strany. </p><p>Pro poloměr kružnice vepsané platí </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 \rho ={\frac {1}{2}}(a+b+c)\operatorname {tg} {\frac {\alpha }{2}}\operatorname {tg} {\frac {\beta }{2}}\operatorname {tg} {\frac {\gamma }{2}}={\frac {S}{s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>tg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mi>tg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>tg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>s</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {1}{2}}(a+b+c)\operatorname {tg} {\frac {\alpha }{2}}\operatorname {tg} {\frac {\beta }{2}}\operatorname {tg} {\frac {\gamma }{2}}={\frac {S}{s}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44374f629efb1ba20cf625dcbd7b121e77b1520f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.218ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {1}{2}}(a+b+c)\operatorname {tg} {\frac {\alpha }{2}}\operatorname {tg} {\frac {\beta }{2}}\operatorname {tg} {\frac {\gamma }{2}}={\frac {S}{s}}.}"></span></dd></dl> <p>Vzdálenost mezi středy kružnice vepsané a opsané je </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 d={\sqrt {r^{2}-2r\rho }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {r^{2}-2r\rho }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/544b2fdef8d06887a96f85276771be491d427d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.641ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {r^{2}-2r\rho }}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kružnice_připsaná"><span id="Kru.C5.BEnice_p.C5.99ipsan.C3.A1"></span>Kružnice připsaná</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=22" title="Editace sekce: Kružnice připsaná" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=22" title="Editovat zdrojový kód sekce Kružnice připsaná"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Kru%C5%BEnice_p%C5%99ipsan%C3%A1" title="Kružnice připsaná">Kružnice připsaná</a> trojúhelníku je kružnice, která se dotýká jedné strany trojúhelníku a dvou přímek, které jsou prodloužením zbývajících stran trojúhelníku. Střed kružnice připsané leží v průsečíku osy jednoho vnitřního úhlu a dvou vedlejších úhlů při zbývajících dvou vrcholech. Každý trojúhelník má tři kružnice připsané. </p> <div class="mw-heading mw-heading3"><h3 id="Kružnice_devíti_bodů"><span id="Kru.C5.BEnice_dev.C3.ADti_bod.C5.AF"></span>Kružnice devíti bodů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=23" title="Editace sekce: Kružnice devíti bodů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=23" title="Editovat zdrojový kód sekce Kružnice devíti bodů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="uvodni-upozorneni hatnote"> Podrobnější informace naleznete v článku <a href="/wiki/Kru%C5%BEnice_dev%C3%ADti_bod%C5%AF" title="Kružnice devíti bodů">Kružnice devíti bodů</a>.</div> <p>Na kružnici devíti bodů leží středy stran, paty výšek a středy spojnic vrcholů se společným průsečíkem výšek. Také se dotýká kružnice vepsané a kružnic připsaných. </p> <div class="mw-heading mw-heading2"><h2 id="Konstrukce_trojúhelníku"><span id="Konstrukce_troj.C3.BAheln.C3.ADku"></span>Konstrukce trojúhelníku</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=24" title="Editace sekce: Konstrukce trojúhelníku" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=24" title="Editovat zdrojový kód sekce Konstrukce trojúhelníku"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="uvodni-upozorneni hatnote"> Podrobnější informace naleznete v článku <a href="/wiki/Konstrukce_troj%C3%BAheln%C3%ADku" title="Konstrukce trojúhelníku">Konstrukce trojúhelníku</a>.</div> <p>Trojúhelník může být určen: </p> <ul><li>(sss) délkou všech tří stran,</li> <li>(sus) délkou dvou stran a velikostí úhlu, který svírají,</li> <li>(usu) délkou strany a velikostí úhlů, které k ní přiléhají,</li> <li>(Ssu) délkou dvou stran a velikostí úhlu proti větší z nich.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <p>Ke konstrukci trojúhelníku se mohou použít i výšky, těžnice atd. </p> <div class="mw-heading mw-heading2"><h2 id="Odkazy">Odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=25" title="Editace sekce: Odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=25" title="Editovat zdrojový kód sekce Odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Literatura">Literatura</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=26" title="Editace sekce: Literatura" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=26" title="Editovat zdrojový kód sekce Literatura"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="book" style="font-style:normal;"><a href="/wiki/Jaroslav_%C5%A0vr%C4%8Dek_(matematik)" title="Jaroslav Švrček (matematik)">ŠVRČEK, Jaroslav</a>; VANŽURA, Jiří. <i>Geometrie trojúhelníka</i>. Praha: Nakladatelství technické literatury, 1988.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Geometrie+troj%C3%BAheln%C3%ADka&rft.aulast=%C5%A0vr%C4%8Dek&rft.aufirst=Jaroslav&rft.au=Van%C5%BEura%2C+Ji%C5%99%C3%AD&rft.place=Praha&rft.pub=Nakladatelstv%C3%AD+technick%C3%A9+literatury&rft.date=1988"><span style="display:none"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Reference">Reference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=27" title="Editace sekce: Reference" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=27" title="Editovat zdrojový kód sekce Reference"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="book" style="font-style:normal;">KRYNICKÝ, Martin. <i>Vnitřní a vnější úhly trojúhelníku I.</i> [online]. Realisticky.cz, 2013-04-03 [cit. 2023-03-06]. <a rel="nofollow" class="external text" href="http://www.realisticky.cz/ucebnice/03%20Matematika%20Z%C5%A0/01%206.%20ro%C4%8Dn%C3%ADk/05%20%C3%9Ahel/15%20Vnit%C5%99n%C3%AD%20a%20vn%C4%9Bj%C5%A1%C3%AD%20%C3%BAhly%20troj%C3%BAheln%C3%ADku%20I.pdf">Dostupné online</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitaceelmonografie&rft.btitle=Vnit%C5%99n%C3%AD+a+vn%C4%9Bj%C5%A1%C3%AD+%C3%BAhly+troj%C3%BAheln%C3%ADku+I.&rft_id=http%3A%2F%2Fwww.realisticky.cz%2Fucebnice%2F03%2520Matematika%2520Z%25C5%25A0%2F01%25206.%2520ro%25C4%258Dn%25C3%25ADk%2F05%2520%25C3%259Ahel%2F15%2520Vnit%25C5%2599n%25C3%25AD%2520a%2520vn%25C4%259Bj%25C5%25A1%25C3%25AD%2520%25C3%25BAhly%2520troj%25C3%25BAheln%25C3%25ADku%2520I.pdf&rft.aulast=Krynick%C3%BD&rft.aufirst=Martin&rft.pub=Realisticky.cz"><span style="display:none"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite style="font-style:normal;"> Konstrukce trojúhelníků: věty sss, sus, usu, Ssu – Procvičování online – Umíme matiku. <i>www.umimematiku.cz</i> [online]. [cit. 2023-06-26]. <a rel="nofollow" class="external text" href="https://www.umimematiku.cz/cviceni-konstrukce-trojuhelniku-vety-sss-sus-usu-ssu">Dostupné online</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceelperiodika&rft.jtitle=www.umimematiku.cz&rft_id=https%3A%2F%2Fwww.umimematiku.cz%2Fcviceni-konstrukce-trojuhelniku-vety-sss-sus-usu-ssu&rft.atitle=Konstrukce+troj%C3%BAheln%C3%ADk%C5%AF%3A+v%C4%9Bty+sss%2C+sus%2C+usu%2C+Ssu+%E2%80%93+Procvi%C4%8Dov%C3%A1n%C3%AD+online+%E2%80%93+Um%C3%ADme+matiku"><span style="display:none"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=28" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=28" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Pravo%C3%BAhl%C3%BD_troj%C3%BAheln%C3%ADk" title="Pravoúhlý trojúhelník">Pravoúhlý trojúhelník</a></li> <li><a href="/wiki/Matematika" title="Matematika">Matematika</a></li> <li><a href="/wiki/Geometrie" title="Geometrie">Geometrie</a></li> <li><a href="/wiki/Planimetrie" title="Planimetrie">Planimetrie</a> – rovinné geometrické útvary</li> <li><a href="/wiki/Geometrick%C3%BD_%C3%BAtvar" title="Geometrický útvar">Geometrický útvar</a></li> <li><a href="/wiki/V%C4%9Bty_o_shodnosti_troj%C3%BAheln%C3%ADku" title="Věty o shodnosti trojúhelníku">Věty o shodnosti trojúhelníku</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&veaction=edit&section=29" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Troj%C3%BAheln%C3%ADk&action=edit&section=29" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons"><img alt="Logo Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://commons.wikimedia.org/wiki/Category:Triangles" class="extiw" title="c:Category:Triangles">trojúhelník</a></span> na <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></span></span></span><i> </i></li> <li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-file-width="411" data-file-height="411" /></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://cs.wiktionary.org/wiki/troj%C3%BAheln%C3%ADk" class="extiw" title="wikt:trojúhelník">trojúhelník</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161228073202/http://trig.wz.cz/">Výpočet úhlů a stran pravoúhlého trojúhelníku</a></li> <li><a rel="nofollow" class="external text" href="http://www.matematika.webz.cz/ostatni/trojuhelnik/">Základní konstrukce v trojúhelníku</a> – flashová animace sestrojení výšek, těžnic, kružnic opsané a vepsané</li></ul> <div class="navbox noprint" title="Mnohoúhelníky"><table title="" class="navbox-navbox mw-collapsible autocollapse nowraplinks noprint" cellspacing="0" data-collapsetext="skrýt" data-expandtext="zobrazit"><style data-mw-deduplicate="TemplateStyles:r17064757">.mw-parser-output .navbox-title .mw-collapsible-toggle{font-weight:normal}</style><tbody><tr> <th class="navbox-title" colspan="2"> <span class="navbox-titletext"><a href="/wiki/Mnoho%C3%BAheln%C3%ADk" title="Mnohoúhelník">Mnohoúhelníky</a></span> </th> </tr><tr class="navbox-padding"><td></td></tr> <tr> <td class="navbox-list navbox-odd" colspan="2"> <div> <a class="mw-selflink selflink">Trojúhelník</a> • <a href="/wiki/%C4%8Cty%C5%99%C3%BAheln%C3%ADk" title="Čtyřúhelník">Čtyřúhelník</a> • <a href="/wiki/P%C4%9Bti%C3%BAheln%C3%ADk" title="Pětiúhelník">Pětiúhelník</a> • <a href="/wiki/%C5%A0esti%C3%BAheln%C3%ADk" title="Šestiúhelník">Šestiúhelník</a> • <a href="/wiki/Sedmi%C3%BAheln%C3%ADk" title="Sedmiúhelník">Sedmiúhelník</a> • <a href="/wiki/Osmi%C3%BAheln%C3%ADk" title="Osmiúhelník">Osmiúhelník</a> • <a href="/wiki/Dev%C3%ADti%C3%BAheln%C3%ADk" title="Devítiúhelník">Devítiúhelník</a> • <a href="/wiki/Deseti%C3%BAheln%C3%ADk" title="Desetiúhelník">Desetiúhelník</a> • <a href="/wiki/Jeden%C3%A1cti%C3%BAheln%C3%ADk" title="Jedenáctiúhelník">Jedenáctiúhelník</a> • <a href="/wiki/Dvan%C3%A1cti%C3%BAheln%C3%ADk" title="Dvanáctiúhelník">Dvanáctiúhelník</a> • <a href="/wiki/T%C5%99in%C3%A1cti%C3%BAheln%C3%ADk" title="Třináctiúhelník">Třináctiúhelník</a> • <a href="/wiki/%C4%8Ctrn%C3%A1cti%C3%BAheln%C3%ADk" title="Čtrnáctiúhelník">Čtrnáctiúhelník</a> • <a href="/wiki/Patn%C3%A1cti%C3%BAheln%C3%ADk" title="Patnáctiúhelník">Patnáctiúhelník</a> • <a href="/wiki/%C5%A0estn%C3%A1cti%C3%BAheln%C3%ADk" title="Šestnáctiúhelník">Šestnáctiúhelník</a> • <a href="/wiki/Sedmn%C3%A1cti%C3%BAheln%C3%ADk" title="Sedmnáctiúhelník">Sedmnáctiúhelník</a> • <a href="/wiki/Osmn%C3%A1cti%C3%BAheln%C3%ADk" title="Osmnáctiúhelník">Osmnáctiúhelník</a> • <a href="/wiki/Devaten%C3%A1cti%C3%BAheln%C3%ADk" title="Devatenáctiúhelník">Devatenáctiúhelník</a> • <a href="/wiki/Dvaceti%C3%BAheln%C3%ADk" title="Dvacetiúhelník">Dvacetiúhelník</a> • <a href="/wiki/Jednadvaceti%C3%BAheln%C3%ADk" title="Jednadvacetiúhelník">Jednadvacetiúhelník</a> • <a href="/wiki/Dvaadvaceti%C3%BAheln%C3%ADk" title="Dvaadvacetiúhelník">Dvaadvacetiúhelník</a> • <a href="/w/index.php?title=%C4%8Cty%C5%99iadvaceti%C3%BAheln%C3%ADk&action=edit&redlink=1" class="new" title="Čtyřiadvacetiúhelník (stránka neexistuje)">Čtyřiadvacetiúhelník</a> • <a href="/wiki/T%C5%99iceti%C3%BAheln%C3%ADk" class="mw-redirect" title="Třicetiúhelník">Třicetiúhelník</a></div> </td> </tr></tbody></table> </div> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output .navbox2{box-sizing:border-box;border:1px solid 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