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ez da ezinbestekoa, ordea." class=""><span>Sortu kontua</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=Berezi:SaioaHasi&returnto=Pitagorasen+teorema" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o" class=""><span>Hasi saioa</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="Aukera gehiago" > <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="Tresna pertsonalak" > <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">Tresna pertsonalak</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User 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_eu.wikipedia.org&uselang=eu"><span>Dohaintza egin</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:KontuaSortu&returnto=Pitagorasen+teorema" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Sortu kontua</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:SaioaHasi&returnto=Pitagorasen+teorema" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Hasi saioa</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"> Izena eman gabeko erabiltzaileentzako orrialdeak <a href="/wiki/Laguntza:Sarrera" aria-label="Artikuluak aldatzeari buruz gehiago ikasi"><span>gehiago ikasi</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/Berezi:NireEkarpenak" title="IP helbide honetatik egindako aldaketen zerrenda [y]" accesskey="y"><span>Ekarpenak</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Berezi:NireEztabaida" title="Zure IParen eztabaida [n]" accesskey="n"><span>Eztabaida</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="Gunea"> <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="Edukiak" 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">Edukiak</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ezkutatu</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">⇑ Gora</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <ul id="toc-Historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frogak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frogak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Frogak</span> </div> </a> <button aria-controls="toc-Frogak-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>Erakutsi/ezkutatu Frogak azpiatal</span> </button> <ul id="toc-Frogak-sublist" class="vector-toc-list"> <li id="toc-Froga_algebraikoak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Froga_algebraikoak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Froga algebraikoak</span> </div> </a> <ul id="toc-Froga_algebraikoak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Froga_geometrikoak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Froga_geometrikoak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Froga geometrikoak</span> </div> </a> <ul id="toc-Froga_geometrikoak-sublist" class="vector-toc-list"> <li id="toc-Triangeluen_antzekotasuna" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Triangeluen_antzekotasuna"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Triangeluen antzekotasuna</span> </div> </a> <ul id="toc-Triangeluen_antzekotasuna-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einsteinen_froga_trigonometrikoa" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Einsteinen_froga_trigonometrikoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Einsteinen froga trigonometrikoa</span> </div> </a> <ul id="toc-Einsteinen_froga_trigonometrikoa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Froga_grafikoa" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Froga_grafikoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Froga grafikoa</span> </div> </a> <ul id="toc-Froga_grafikoa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Froga_analitikoa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Froga_analitikoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Froga analitikoa</span> </div> </a> <ul id="toc-Froga_analitikoa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teoremaren_alderantzizkoa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teoremaren_alderantzizkoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Teoremaren alderantzizkoa</span> </div> </a> <ul id="toc-Teoremaren_alderantzizkoa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Erabilera_adibideak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erabilera_adibideak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Erabilera adibideak</span> </div> </a> <button aria-controls="toc-Erabilera_adibideak-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>Erakutsi/ezkutatu Erabilera adibideak azpiatal</span> </button> <ul id="toc-Erabilera_adibideak-sublist" class="vector-toc-list"> <li id="toc-Hirukote_pitagorikoak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hirukote_pitagorikoak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hirukote pitagorikoak</span> </div> </a> <ul id="toc-Hirukote_pitagorikoak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zenbaki_neurtezinak_eraikitzea" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zenbaki_neurtezinak_eraikitzea"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Zenbaki neurtezinak eraikitzea</span> </div> </a> <ul id="toc-Zenbaki_neurtezinak_eraikitzea-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zenbaki_konplexuak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zenbaki_konplexuak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Zenbaki konplexuak</span> </div> </a> <ul id="toc-Zenbaki_konplexuak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometria_euklidearra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometria_euklidearra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Geometria euklidearra</span> </div> </a> <ul id="toc-Geometria_euklidearra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pitagorasen_identitate_trigonometrikoa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pitagorasen_identitate_trigonometrikoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Pitagorasen identitate trigonometrikoa</span> </div> </a> <ul id="toc-Pitagorasen_identitate_trigonometrikoa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biderketa_bektoriala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biderketa_bektoriala"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Biderketa bektoriala</span> </div> </a> <ul id="toc-Biderketa_bektoriala-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pitagorasen_alderantzizko_teorema" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pitagorasen_alderantzizko_teorema"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Pitagorasen alderantzizko teorema</span> </div> </a> <ul id="toc-Pitagorasen_alderantzizko_teorema-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gehiago" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gehiago"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Gehiago</span> </div> </a> <ul id="toc-Gehiago-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orokortzeak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orokortzeak"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Orokortzeak</span> </div> </a> <button aria-controls="toc-Orokortzeak-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>Erakutsi/ezkutatu Orokortzeak azpiatal</span> </button> <ul id="toc-Orokortzeak-sublist" class="vector-toc-list"> <li id="toc-Kosinuaren_teorema" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kosinuaren_teorema"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Kosinuaren teorema</span> </div> </a> <ul id="toc-Kosinuaren_teorema-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Barne_produktudun_espazioak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Barne_produktudun_espazioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Barne produktudun espazioak</span> </div> </a> <ul id="toc-Barne_produktudun_espazioak-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ariketak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ariketak"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span><figcaption></figcaption><p> Ariketak</p></span> </div> </a> <ul id="toc-Ariketak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Erreferentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erreferentziak"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Erreferentziak</span> </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Kanpo estekak</span> </div> </a> <ul id="toc-Kanpo_estekak-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Edukiak" 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="Eduki taularen ikusgarritasuna aldatu" > <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">Eduki taularen ikusgarritasuna aldatu</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">Pitagorasen teorema</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="Joan beste hizkuntza batean idatzitako artikulu batera. 132 hizkuntzatan eskuragarri." > <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-132" 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">132 hizkuntza</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Pythagoras_se_stelling" title="Pythagoras se stelling – afrikaansa" lang="af" hreflang="af" data-title="Pythagoras se stelling" data-language-autonym="Afrikaans" data-language-local-name="afrikaansa" 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/Satz_des_Pythagoras" title="Satz des Pythagoras – Suitzako alemana" lang="gsw" hreflang="gsw" data-title="Satz des Pythagoras" data-language-autonym="Alemannisch" data-language-local-name="Suitzako alemana" 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%8D%93%E1%8B%AD%E1%89%B3%E1%8C%8E%E1%88%A8%E1%88%B5_%E1%8A%A5%E1%88%AD%E1%8C%89%E1%8C%A5" title="ፓይታጎረስ እርጉጥ – amharera" lang="am" hreflang="am" data-title="ፓይታጎረስ እርጉጥ" data-language-autonym="አማርኛ" data-language-local-name="amharera" 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/Teorema_de_Pitagoras" title="Teorema de Pitagoras – aragoiera" lang="an" hreflang="an" data-title="Teorema de Pitagoras" data-language-autonym="Aragonés" data-language-local-name="aragoiera" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D9%81%D9%8A%D8%AB%D8%A7%D8%BA%D9%88%D8%B1%D8%B3" title="مبرهنة فيثاغورس – arabiera" lang="ar" hreflang="ar" data-title="مبرهنة فيثاغورس" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%87%E0%A6%A5%E0%A6%BE%E0%A6%97%E0%A7%8B%E0%A7%B0%E0%A6%BE%E0%A6%9B%E0%A7%B0_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="পাইথাগোৰাছৰ উপপাদ্য – assamera" lang="as" hreflang="as" data-title="পাইথাগোৰাছৰ উপপাদ্য" data-language-autonym="অসমীয়া" data-language-local-name="assamera" 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/Teorema_de_Pit%C3%A1gores" title="Teorema de Pitágores – asturiera" lang="ast" hreflang="ast" data-title="Teorema de Pitágores" data-language-autonym="Asturianu" data-language-local-name="asturiera" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%87%E0%A4%A5%E0%A4%BE%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%B8_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="पाइथागोरस प्रमेय – awadhiera" lang="awa" hreflang="awa" data-title="पाइथागोरस प्रमेय" data-language-autonym="अवधी" data-language-local-name="awadhiera" class="interlanguage-link-target"><span>अवधी</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Pifaqor_teoremi" title="Pifaqor teoremi – azerbaijanera" lang="az" hreflang="az" data-title="Pifaqor teoremi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" 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/%D9%81%DB%8C%D8%AB%D8%A7%D8%BA%D9%88%D8%B1%D8%B3_%D8%AA%D8%A6%D9%88%D8%B1%DB%8C%D8%B3%DB%8C" title="فیثاغورس تئوریسی – South Azerbaijani" lang="azb" hreflang="azb" data-title="فیثاغورس تئوریسی" 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/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Пифагор теоремаһы – baxkirera" lang="ba" hreflang="ba" data-title="Пифагор теоремаһы" data-language-autonym="Башҡортса" data-language-local-name="baxkirera" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/Rumus_Pythagoras" title="Rumus Pythagoras – baliera" lang="ban" hreflang="ban" data-title="Rumus Pythagoras" data-language-autonym="Basa Bali" data-language-local-name="baliera" class="interlanguage-link-target"><span>Basa Bali</span></a></li><li class="interlanguage-link interwiki-bar badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://bar.wikipedia.org/wiki/Sotz_vum_Pythagoras" title="Sotz vum Pythagoras – Bavarian" lang="bar" hreflang="bar" data-title="Sotz vum Pythagoras" data-language-autonym="Boarisch" data-language-local-name="Bavarian" 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/P%C4%97taguora_teuorema" title="Pėtaguora teuorema – Samogitian" lang="sgs" hreflang="sgs" data-title="Pėtaguora teuorema" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" 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/Teorema_ni_Pythagoras" title="Teorema ni Pythagoras – Central Bikol" lang="bcl" hreflang="bcl" data-title="Teorema ni Pythagoras" 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%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0_%D0%9F%D1%96%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D0%B0" title="Тэарэма Піфагора – bielorrusiera" lang="be" hreflang="be" data-title="Тэарэма Піфагора" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" 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%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0_%D0%9F%D1%96%D1%82%D0%B0%D0%B3%D0%BE%D1%80%D0%B0" title="Тэарэма Пітагора – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэарэма Пітагора" 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%9F%D0%B8%D1%82%D0%B0%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Питагорова теорема – bulgariera" lang="bg" hreflang="bg" data-title="Питагорова теорема" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BF%E0%A6%A5%E0%A6%BE%E0%A6%97%E0%A7%8B%E0%A6%B0%E0%A6%BE%E0%A6%B8%E0%A7%87%E0%A6%B0_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="পিথাগোরাসের উপপাদ্য – bengalera" lang="bn" hreflang="bn" data-title="পিথাগোরাসের উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Teorem_Pythagoras" title="Teorem Pythagoras – bretoiera" lang="br" hreflang="br" data-title="Teorem Pythagoras" data-language-autonym="Brezhoneg" data-language-local-name="bretoiera" 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/Pitagorina_teorema" title="Pitagorina teorema – bosniera" lang="bs" hreflang="bs" data-title="Pitagorina teorema" data-language-autonym="Bosanski" data-language-local-name="bosniera" 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/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores – katalana" lang="ca" hreflang="ca" data-title="Teorema de Pitàgores" data-language-autonym="Català" data-language-local-name="katalana" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%D9%BE%DB%8C%D8%AA%D8%A7%DA%AF%DB%86%D8%B1%D8%B3" title="تیۆرمی پیتاگۆرس – erdialdeko kurduera" lang="ckb" hreflang="ckb" data-title="تیۆرمی پیتاگۆرس" data-language-autonym="کوردی" data-language-local-name="erdialdeko kurduera" 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/Tiurema_di_Pitagora" title="Tiurema di Pitagora – korsikera" lang="co" hreflang="co" data-title="Tiurema di Pitagora" data-language-autonym="Corsu" data-language-local-name="korsikera" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pythagorova_v%C4%9Bta" title="Pythagorova věta – txekiera" lang="cs" hreflang="cs" data-title="Pythagorova věta" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B8" title="Пифагор теореми – txuvaxera" lang="cv" hreflang="cv" data-title="Пифагор теореми" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" 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/Theorem_Pythagoras" title="Theorem Pythagoras – galesa" lang="cy" hreflang="cy" data-title="Theorem Pythagoras" data-language-autonym="Cymraeg" data-language-local-name="galesa" 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/Den_pythagor%C3%A6iske_l%C3%A6res%C3%A6tning" title="Den pythagoræiske læresætning – daniera" lang="da" hreflang="da" data-title="Den pythagoræiske læresætning" data-language-autonym="Dansk" data-language-local-name="daniera" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://de.wikipedia.org/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras – alemana" lang="de" hreflang="de" data-title="Satz des Pythagoras" data-language-autonym="Deutsch" data-language-local-name="alemana" 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/Teorem%C3%AA_Pisagori" title="Teoremê Pisagori – Zazaki" lang="diq" hreflang="diq" data-title="Teoremê Pisagori" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Teorem_Pythagoras" title="Teorem Pythagoras – Central Dusun" lang="dtp" hreflang="dtp" data-title="Teorem Pythagoras" data-language-autonym="Kadazandusun" data-language-local-name="Central Dusun" class="interlanguage-link-target"><span>Kadazandusun</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%85%CE%B8%CE%B1%CE%B3%CF%8C%CF%81%CE%B5%CE%B9%CE%BF_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Πυθαγόρειο θεώρημα – greziera" lang="el" hreflang="el" data-title="Πυθαγόρειο θεώρημα" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Tior%C3%A9ma_%27d_Pit%C3%A0gora" title="Tioréma 'd Pitàgora – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Tioréma 'd Pitàgora" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem – ingelesa" lang="en" hreflang="en" data-title="Pythagorean theorem" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://eo.wikipedia.org/wiki/Teoremo_de_Pitagoro" title="Teoremo de Pitagoro – esperantoa" lang="eo" hreflang="eo" data-title="Teoremo de Pitagoro" data-language-autonym="Esperanto" data-language-local-name="esperantoa" 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/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras – gaztelania" lang="es" hreflang="es" data-title="Teorema de Pitágoras" data-language-autonym="Español" data-language-local-name="gaztelania" 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/Pythagorase_teoreem" title="Pythagorase teoreem – estoniera" lang="et" hreflang="et" data-title="Pythagorase teoreem" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D9%81%DB%8C%D8%AB%D8%A7%D8%BA%D9%88%D8%B1%D8%B3" title="قضیه فیثاغورس – persiera" lang="fa" hreflang="fa" data-title="قضیه فیثاغورس" data-language-autonym="فارسی" data-language-local-name="persiera" 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/Pythagoraan_lause" title="Pythagoraan lause – finlandiera" lang="fi" hreflang="fi" data-title="Pythagoraan lause" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Pythagore" title="Théorème de Pythagore – frantsesa" lang="fr" hreflang="fr" data-title="Théorème de Pythagore" data-language-autonym="Français" data-language-local-name="frantsesa" 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/Pythagoras_sin_reegel" title="Pythagoras sin reegel – iparraldeko frisiera" lang="frr" hreflang="frr" data-title="Pythagoras sin reegel" data-language-autonym="Nordfriisk" data-language-local-name="iparraldeko frisiera" 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/Teoirim_Ph%C3%ADotagar%C3%A1sach" title="Teoirim Phíotagarásach – irlandera" lang="ga" hreflang="ga" data-title="Teoirim Phíotagarásach" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras – galiziera" lang="gl" hreflang="gl" data-title="Teorema de Pitágoras" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Pit%C3%A1gora_remimoa%C3%B1eter%C3%A3" title="Pitágora remimoañeterã – guaraniera" lang="gn" hreflang="gn" data-title="Pitágora remimoañeterã" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraniera" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AA%BE%E0%AA%AF%E0%AA%A5%E0%AA%BE%E0%AA%97%E0%AB%8B%E0%AA%B0%E0%AA%B8%E0%AA%A8%E0%AB%81%E0%AA%82_%E0%AA%AA%E0%AB%8D%E0%AA%B0%E0%AA%AE%E0%AB%87%E0%AA%AF" title="પાયથાગોરસનું પ્રમેય – gujaratera" lang="gu" hreflang="gu" data-title="પાયથાગોરસનું પ્રમેય" data-language-autonym="ગુજરાતી" data-language-local-name="gujaratera" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A4%D7%99%D7%AA%D7%92%D7%95%D7%A8%D7%A1" title="משפט פיתגורס – hebreera" lang="he" hreflang="he" data-title="משפט פיתגורס" data-language-autonym="עברית" data-language-local-name="hebreera" 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%AA%E0%A4%BE%E0%A4%87%E0%A4%A5%E0%A4%BE%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%B8_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="पाइथागोरस प्रमेय – hindia" lang="hi" hreflang="hi" data-title="पाइथागोरस प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem – Fiji Hindi" lang="hif" hreflang="hif" data-title="Pythagorean theorem" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pitagorin_pou%C4%8Dak" title="Pitagorin poučak – kroaziera" lang="hr" hreflang="hr" data-title="Pitagorin poučak" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" 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/Sada_Pythagorasa" title="Sada Pythagorasa – goi-sorabiera" lang="hsb" hreflang="hsb" data-title="Sada Pythagorasa" data-language-autonym="Hornjoserbsce" data-language-local-name="goi-sorabiera" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pitagorasz-t%C3%A9tel" title="Pitagorasz-tétel – hungariera" lang="hu" hreflang="hu" data-title="Pitagorasz-tétel" data-language-autonym="Magyar" data-language-local-name="hungariera" 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/%D5%8A%D5%B5%D5%B8%D6%82%D5%A9%D5%A1%D5%A3%D5%B8%D6%80%D5%A1%D5%BD%D5%AB_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Պյութագորասի թեորեմ – armeniera" lang="hy" hreflang="hy" data-title="Պյութագորասի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="armeniera" 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/Theorema_de_Pythagoras" title="Theorema de Pythagoras – interlingua" lang="ia" hreflang="ia" data-title="Theorema de Pythagoras" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Teorem_Pythagoras" title="Teorem Pythagoras – ibanera" lang="iba" hreflang="iba" data-title="Teorem Pythagoras" data-language-autonym="Jaku Iban" data-language-local-name="ibanera" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_Pythagoras" title="Teorema Pythagoras – indonesiera" lang="id" hreflang="id" data-title="Teorema Pythagoras" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" 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/Teoremo_di_Pitagoro" title="Teoremo di Pitagoro – idoa" lang="io" hreflang="io" data-title="Teoremo di Pitagoro" data-language-autonym="Ido" data-language-local-name="idoa" 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/Regla_P%C3%BD%C3%BEag%C3%B3rasar" title="Regla Pýþagórasar – islandiera" lang="is" hreflang="is" data-title="Regla Pýþagórasar" data-language-autonym="Íslenska" data-language-local-name="islandiera" 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/Teorema_di_Pitagora" title="Teorema di Pitagora – italiera" lang="it" hreflang="it" data-title="Teorema di Pitagora" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%94%E3%82%BF%E3%82%B4%E3%83%A9%E3%82%B9%E3%81%AE%E5%AE%9A%E7%90%86" title="ピタゴラスの定理 – japoniera" lang="ja" hreflang="ja" data-title="ピタゴラスの定理" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%98%E1%83%97%E1%83%90%E1%83%92%E1%83%9D%E1%83%A0%E1%83%90%E1%83%A1_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%90" title="პითაგორას თეორემა – georgiera" lang="ka" hreflang="ka" data-title="პითაგორას თეორემა" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Asekkud_n_Pythagore" title="Asekkud n Pythagore – kabiliera" lang="kab" hreflang="kab" data-title="Asekkud n Pythagore" data-language-autonym="Taqbaylit" data-language-local-name="kabiliera" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D0%B8_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%8D" title="Пифагор и теоремэ – kabardiera" lang="kbd" hreflang="kbd" data-title="Пифагор и теоремэ" data-language-autonym="Адыгэбзэ" data-language-local-name="kabardiera" 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/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Пифагор теоремасы – kazakhera" lang="kk" hreflang="kk" data-title="Пифагор теоремасы" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%91%E1%9F%92%E1%9E%9A%E1%9E%B9%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%B8%E1%9E%94%E1%9E%91%E1%9E%96%E1%9E%B8%E1%9E%8F%E1%9E%B6%E1%9E%80%E1%9E%9A" title="ទ្រឹស្តីបទពីតាករ – khemerera" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទពីតាករ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khemerera" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%BC%ED%83%80%EA%B3%A0%EB%9D%BC%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="피타고라스 정리 – koreera" lang="ko" hreflang="ko" data-title="피타고라스 정리" data-language-autonym="한국어" data-language-local-name="koreera" 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/Teorema_P%C3%AEtagoras" title="Teorema Pîtagoras – kurduera" lang="ku" hreflang="ku" data-title="Teorema Pîtagoras" data-language-autonym="Kurdî" data-language-local-name="kurduera" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Пифагор теоремасы – kirgizera" lang="ky" hreflang="ky" data-title="Пифагор теоремасы" data-language-autonym="Кыргызча" data-language-local-name="kirgizera" 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/Theorema_Pythagorae" title="Theorema Pythagorae – latina" lang="la" hreflang="la" data-title="Theorema Pythagorae" 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/Teorem_de_Pitagora" title="Teorem de Pitagora – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Teorem de Pitagora" 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-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_de_Pitagora" title="Teorema de Pitagora – Lombard" lang="lmo" hreflang="lmo" data-title="Teorema de Pitagora" data-language-autonym="Lombard" data-language-local-name="Lombard" 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/Bondeko_ya_mpanzi-mis%C3%A1to" title="Bondeko ya mpanzi-misáto – lingala" lang="ln" hreflang="ln" data-title="Bondeko ya mpanzi-misáto" data-language-autonym="Lingála" data-language-local-name="lingala" 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%97%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B0%E0%BA%94%E0%BA%B5_%E0%BA%9B%E0%BA%B5%E0%BA%97%E0%BA%B2%E0%BB%82%E0%BA%81%E0%BA%A3%E0%BA%BD%E0%BA%99" title="ທິດສະດີ ປີທາໂກຣຽນ – laosera" lang="lo" hreflang="lo" data-title="ທິດສະດີ ປີທາໂກຣຽນ" data-language-autonym="ລາວ" data-language-local-name="laosera" 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/Pitagoro_teorema" title="Pitagoro teorema – lituaniera" lang="lt" hreflang="lt" data-title="Pitagoro teorema" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" 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/Pitagora_teor%C4%93ma" title="Pitagora teorēma – letoniera" lang="lv" hreflang="lv" data-title="Pitagora teorēma" data-language-autonym="Latviešu" data-language-local-name="letoniera" 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/Raikitr%27_Pytagore" title="Raikitr' Pytagore – malgaxea" lang="mg" hreflang="mg" data-title="Raikitr' Pytagore" data-language-autonym="Malagasy" data-language-local-name="malgaxea" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%B8%D1%82%D0%B0%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Питагорова теорема – mazedoniera" lang="mk" hreflang="mk" data-title="Питагорова теорема" data-language-autonym="Македонски" data-language-local-name="mazedoniera" 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%AA%E0%B5%88%E0%B4%A4%E0%B4%97%E0%B5%8B%E0%B4%B1%E0%B4%B8%E0%B5%8D_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="പൈതഗോറസ് സിദ്ധാന്തം – malabarera" lang="ml" hreflang="ml" data-title="പൈതഗോറസ് സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="malabarera" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D1%8B%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC" title="Пифагорын теорем – mongoliera" lang="mn" hreflang="mn" data-title="Пифагорын теорем" data-language-autonym="Монгол" data-language-local-name="mongoliera" 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%AA%E0%A4%BE%E0%A4%AF%E0%A4%A5%E0%A4%BE%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%B8%E0%A4%9A%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="पायथागोरसचा सिद्धान्त – marathera" lang="mr" hreflang="mr" data-title="पायथागोरसचा सिद्धान्त" data-language-autonym="मराठी" data-language-local-name="marathera" 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/Teorem_Pythagoras" title="Teorem Pythagoras – malaysiera" lang="ms" hreflang="ms" data-title="Teorem Pythagoras" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%95%E1%80%AD%E1%80%AF%E1%80%80%E1%80%BA%E1%80%9E%E1%80%AC%E1%80%82%E1%80%AD%E1%80%AF%E1%80%9B_%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%99%E1%80%BA" title="ပိုက်သာဂိုရ သီအိုရမ် – birmaniera" lang="my" hreflang="my" data-title="ပိုက်သာဂိုရ သီအိုရမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmaniera" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Satz_van_Pythagoras" title="Satz van Pythagoras – behe-alemana" lang="nds" hreflang="nds" data-title="Satz van Pythagoras" data-language-autonym="Plattdüütsch" data-language-local-name="behe-alemana" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%87%E0%A4%A5%E0%A4%BE%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%B8_%E0%A4%B8%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF" title="पाइथागोरस साध्य – nepalera" lang="ne" hreflang="ne" data-title="पाइथागोरस साध्य" data-language-autonym="नेपाली" data-language-local-name="nepalera" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Pythagoras" title="Stelling van Pythagoras – nederlandera" lang="nl" hreflang="nl" data-title="Stelling van Pythagoras" data-language-autonym="Nederlands" data-language-local-name="nederlandera" 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/Den_pytagoreiske_l%C3%A6resetninga" title="Den pytagoreiske læresetninga – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Den pytagoreiske læresetninga" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://no.wikipedia.org/wiki/Pytagoras%E2%80%99_l%C3%A6resetning" title="Pytagoras’ læresetning – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Pytagoras’ læresetning" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teor%C3%A8ma_de_Pitag%C3%B2ras" title="Teorèma de Pitagòras – okzitaniera" lang="oc" hreflang="oc" data-title="Teorèma de Pitagòras" data-language-autonym="Occitan" data-language-local-name="okzitaniera" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D1%8B_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%C3%A6" title="Пифагоры теоремæ – osetiera" lang="os" hreflang="os" data-title="Пифагоры теоремæ" data-language-autonym="Ирон" data-language-local-name="osetiera" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A8%BE%E0%A8%88%E0%A8%A5%E0%A8%BE%E0%A8%97%E0%A9%8B%E0%A8%B0%E0%A8%B8_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A8%AE" title="ਪਾਈਥਾਗੋਰਸ ਥਿਊਰਮ – punjabera" lang="pa" hreflang="pa" data-title="ਪਾਈਥਾਗੋਰਸ ਥਿਊਰਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabera" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Pitagorasa" title="Twierdzenie Pitagorasa – poloniera" lang="pl" hreflang="pl" data-title="Twierdzenie Pitagorasa" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Pit%C3%A0gora" title="Teorema ëd Pitàgora – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema ëd Pitàgora" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%B3%D9%84%DB%81_%D9%81%DB%8C%D8%B3%D8%A7%D8%BA%D9%88%D8%B1%D8%B3" title="مسلہ فیساغورس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مسلہ فیساغورس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras – portugesa" lang="pt" hreflang="pt" data-title="Teorema de Pitágoras" data-language-autonym="Português" data-language-local-name="portugesa" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-pwn mw-list-item"><a href="https://pwn.wikipedia.org/wiki/sasusuan_ni_pitaguras" title="sasusuan ni pitaguras – Paiwan" lang="pwn" hreflang="pwn" data-title="sasusuan ni pitaguras" data-language-autonym="Pinayuanan" data-language-local-name="Paiwan" class="interlanguage-link-target"><span>Pinayuanan</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_lui_Pitagora" title="Teorema lui Pitagora – errumaniera" lang="ro" hreflang="ro" data-title="Teorema lui Pitagora" data-language-autonym="Română" data-language-local-name="errumaniera" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D0%B0" title="Теорема Пифагора – errusiera" lang="ru" hreflang="ru" data-title="Теорема Пифагора" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%82%D0%B0" title="Пифагор теоремата – sakhera" lang="sah" hreflang="sah" data-title="Пифагор теоремата" data-language-autonym="Саха тыла" data-language-local-name="sakhera" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AF%E1%B1%9F%E1%B1%AD%E1%B1%9B%E1%B1%B7%E1%B1%B3%E1%B1%9C%E1%B1%B3%E1%B1%A8%E1%B1%9A%E1%B1%B1_%E1%B1%9B%E1%B1%B7%E1%B1%A4%E1%B1%AD%E1%B1%9A%E1%B1%A8%E1%B1%9A%E1%B1%A2" title="ᱯᱟᱭᱛᱷᱳᱜᱳᱨᱚᱱ ᱛᱷᱤᱭᱚᱨᱚᱢ – santalera" lang="sat" hreflang="sat" data-title="ᱯᱟᱭᱛᱷᱳᱜᱳᱨᱚᱱ ᱛᱷᱤᱭᱚᱨᱚᱢ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santalera" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiurema_di_Pitagora" title="Tiurema di Pitagora – siziliera" lang="scn" hreflang="scn" data-title="Tiurema di Pitagora" data-language-autonym="Sicilianu" data-language-local-name="siziliera" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Pythagorasa_cealkka" title="Pythagorasa cealkka – iparraldeko samiera" lang="se" hreflang="se" data-title="Pythagorasa cealkka" data-language-autonym="Davvisámegiella" data-language-local-name="iparraldeko samiera" 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/Pitagorina_teorema" title="Pitagorina teorema – serbokroaziera" lang="sh" hreflang="sh" data-title="Pitagorina teorema" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" 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%B4%E0%B6%BA%E0%B7%92%E0%B6%AD%E0%B6%9C%E0%B6%BB%E0%B7%83%E0%B7%8A_%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%9A%E0%B6%BA%E0%B6%BA" title="පයිතගරස් ප්‍රමේයය – sinhala" lang="si" hreflang="si" data-title="පයිතගරස් ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="sinhala" 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/Pythagorean_theorem" title="Pythagorean theorem – Simple English" lang="en-simple" hreflang="en-simple" data-title="Pythagorean theorem" 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/Pytagorova_veta" title="Pytagorova veta – eslovakiera" lang="sk" hreflang="sk" data-title="Pytagorova veta" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" 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/Pitagorov_izrek" title="Pitagorov izrek – esloveniera" lang="sl" hreflang="sl" data-title="Pitagorov izrek" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Dudzirazivo_raPythagoras" title="Dudzirazivo raPythagoras – shonera" lang="sn" hreflang="sn" data-title="Dudzirazivo raPythagoras" data-language-autonym="ChiShona" data-language-local-name="shonera" 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/Aragtida_Baytagoras" title="Aragtida Baytagoras – somaliera" lang="so" hreflang="so" data-title="Aragtida Baytagoras" data-language-autonym="Soomaaliga" data-language-local-name="somaliera" 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/Teorema_e_Pitagor%C3%ABs" title="Teorema e Pitagorës – albaniera" lang="sq" hreflang="sq" data-title="Teorema e Pitagorës" data-language-autonym="Shqip" data-language-local-name="albaniera" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B8%D1%82%D0%B0%D0%B3%D0%BE%D1%80%D0%B8%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Питагорина теорема – serbiera" lang="sr" hreflang="sr" data-title="Питагорина теорема" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pythagoras_sats" title="Pythagoras sats – suediera" lang="sv" hreflang="sv" data-title="Pythagoras sats" data-language-autonym="Svenska" data-language-local-name="suediera" 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/Uhakiki_wa_Pythagoras" title="Uhakiki wa Pythagoras – swahilia" lang="sw" hreflang="sw" data-title="Uhakiki wa Pythagoras" data-language-autonym="Kiswahili" data-language-local-name="swahilia" 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/Twjerdzy%C5%84y_Pitagorasa" title="Twjerdzyńy Pitagorasa – Silesian" lang="szl" hreflang="szl" data-title="Twjerdzyńy Pitagorasa" data-language-autonym="Ślůnski" data-language-local-name="Silesian" 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%AA%E0%AE%BF%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AF%87%E0%AE%95%E0%AF%8B%E0%AE%B0%E0%AE%9A%E0%AF%81_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="பித்தேகோரசு தேற்றம் – tamilera" lang="ta" hreflang="ta" data-title="பித்தேகோரசு தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamilera" 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%AA%E0%B1%88%E0%B0%A5%E0%B0%BE%E0%B0%97%E0%B0%B0%E0%B0%B8%E0%B1%8D_%E0%B0%B8%E0%B0%BF%E0%B0%A6%E0%B1%8D%E0%B0%A7%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B0%82" title="పైథాగరస్ సిద్ధాంతం – telugua" lang="te" hreflang="te" data-title="పైథాగరస్ సిద్ధాంతం" data-language-autonym="తెలుగు" data-language-local-name="telugua" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%9E%E0%B8%B5%E0%B8%97%E0%B8%B2%E0%B9%82%E0%B8%81%E0%B8%A3%E0%B8%B1%E0%B8%AA" title="ทฤษฎีบทพีทาโกรัส – thailandiera" lang="th" hreflang="th" data-title="ทฤษฎีบทพีทาโกรัส" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Pifagory%C5%88_teoremasy" title="Pifagoryň teoremasy – turkmenera" lang="tk" hreflang="tk" data-title="Pifagoryň teoremasy" data-language-autonym="Türkmençe" data-language-local-name="turkmenera" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorema_ni_Pitagoras" title="Teorema ni Pitagoras – tagaloa" lang="tl" hreflang="tl" data-title="Teorema ni Pitagoras" data-language-autonym="Tagalog" data-language-local-name="tagaloa" 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/Pisagor_teoremi" title="Pisagor teoremi – turkiera" lang="tr" hreflang="tr" data-title="Pisagor teoremi" data-language-autonym="Türkçe" data-language-local-name="turkiera" 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/Pifagor_teoremas%C4%B1" title="Pifagor teoreması – tatarera" lang="tt" hreflang="tt" data-title="Pifagor teoreması" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarera" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%96%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D0%B0" title="Теорема Піфагора – ukrainera" lang="uk" hreflang="uk" data-title="Теорема Піфагора" data-language-autonym="Українська" data-language-local-name="ukrainera" 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%B3%D8%A6%D9%84%DB%82_%D9%81%DB%8C%D8%AB%D8%A7_%D8%BA%D9%88%D8%B1%D8%AB" title="مسئلۂ فیثا غورث – urdua" lang="ur" hreflang="ur" data-title="مسئلۂ فیثا غورث" data-language-autonym="اردو" data-language-local-name="urdua" 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/Pifagor_teoremasi" title="Pifagor teoremasi – uzbekera" lang="uz" hreflang="uz" data-title="Pifagor teoremasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekera" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Pifagoran_teorem" title="Pifagoran teorem – Veps" lang="vep" hreflang="vep" data-title="Pifagoran teorem" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Pythagoras" title="Định lý Pythagoras – vietnamera" lang="vi" hreflang="vi" data-title="Định lý Pythagoras" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pitagorasnon_nga_teyorema" title="Pitagorasnon nga teyorema – warayera" lang="war" hreflang="war" data-title="Pitagorasnon nga teyorema" data-language-autonym="Winaray" data-language-local-name="warayera" 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/%E5%8B%BE%E8%82%A1%E5%AE%9A%E7%90%86" title="勾股定理 – wu txinera" lang="wuu" hreflang="wuu" data-title="勾股定理" data-language-autonym="吴语" data-language-local-name="wu txinera" 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%A4%D7%99%D7%98%D7%90%D7%92%D7%90%D7%A8%D7%90%D7%A1_%D7%A4%D7%A8%D7%99%D7%A0%D7%A6%D7%99%D7%A4" title="פיטאגאראס פרינציפ – yiddisha" lang="yi" hreflang="yi" data-title="פיטאגאראס פרינציפ" data-language-autonym="ייִדיש" data-language-local-name="yiddisha" 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/%C3%80gb%C3%A9r%C3%B2_Pythagoras" title="Àgbérò Pythagoras – jorubera" lang="yo" hreflang="yo" data-title="Àgbérò Pythagoras" data-language-autonym="Yorùbá" data-language-local-name="jorubera" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8B%BE%E8%82%A1%E5%AE%9A%E7%90%86" title="勾股定理 – txinera" lang="zh" hreflang="zh" data-title="勾股定理" data-language-autonym="中文" data-language-local-name="txinera" 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/%E5%8B%BE%E8%82%A1%E5%AE%9A%E7%90%86" title="勾股定理 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="勾股定理" data-language-autonym="文言" data-language-local-name="Literary Chinese" 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/Pythagoras_t%C4%93ng-l%C3%AD" title="Pythagoras tēng-lí – Minnan" lang="nan" hreflang="nan" data-title="Pythagoras tēng-lí" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="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/%E7%95%A2%E6%B0%8F%E5%AE%9A%E7%90%86" title="畢氏定理 – kantonera" lang="yue" hreflang="yue" data-title="畢氏定理" data-language-autonym="粵語" data-language-local-name="kantonera" 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/Q11518#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</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="Izen-tarteak"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Pitagorasen_teorema" title="Eduki orrialdea ikusi [c]" accesskey="c"><span>Artikulua</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Eztabaida:Pitagorasen_teorema" rel="discussion" title="Artikuluari buruzko eztabaida [t]" accesskey="t"><span>Eztabaida</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" 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id="mwe_player_2" poster="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Pitagorasen_Teorema_azalpena.webm/220px--Pitagorasen_Teorema_azalpena.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="121" data-mwtitle="Pitagorasen_Teorema_azalpena.webm" data-mwprovider="wikimediacommons" resource="/wiki/Fitxategi:Pitagorasen_Teorema_azalpena.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/d/d1/Pitagorasen_Teorema_azalpena.webm" type="video/webm; codecs="vp9, opus"" data-width="3840" data-height="2160" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/Pitagorasen_Teorema_azalpena.webm/Pitagorasen_Teorema_azalpena.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption><b>Pitagorasen teorema</b> ulertzeko bideoa.<hr /><span style="color: #777;"><figure class="mw-halign-left" typeof="mw:File"><a href="https://www.jakindun.com/" rel="nofollow"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/33px-Jakindun_logoa.png" decoding="async" width="33" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/50px-Jakindun_logoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/66px-Jakindun_logoa.png 2x" data-file-width="225" data-file-height="225" /></a><figcaption></figcaption></figure>Bideo hau Jakindun elkarteak egin du. Gehiago dituzu eskuragarri <a rel="nofollow" class="external text" href="https://www.jakindun.com/">euren gunean</a>. Bideoak dituzten artikulu guztiak ikus ditzakezu <a href="/wiki/Kategoria:Jakindunen_bideoak_dituzten_artikuluak" title="Kategoria:Jakindunen bideoak dituzten artikuluak"><b>hemen</b></a>.</span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pythagorean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/220px-Pythagorean.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/330px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/440px-Pythagorean.svg.png 2x" data-file-width="512" data-file-height="466" /></a><figcaption>Triangelu zuzena. Katetoak <i>a</i> eta <i>b</i> bezala adierazten dira eta hipotenusa <i>c</i> bezala. Bakoitzaren karratuak kolore ezberdinez adierazten dira hemen: <i>a</i><sup>2</sup> (urdina), <i>b</i><sup>2</sup> (gorria), eta <i>c</i><sup>2</sup> (morea).</figcaption></figure> <p><a href="/wiki/Matematika" title="Matematika">Matematikan</a>, <b>Pitagorasen teorema</b> <a href="/wiki/Geometria_euklidear" title="Geometria euklidear">geometria euklidiarrean</a> <a href="/wiki/Triangelu_angeluzuzen" title="Triangelu angeluzuzen">triangelu angeluzuzuen</a> baten hiru aldeen arteko funtsezko erlazioa da. Honakoa dio:<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> </p> <blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: #FFFFFF; color: #000000"> <p><b>Pitagorasen teorema</b></p> <p>Triangelu angeluzuzen batean, katetoen karratuen batura hipotenusaren karratuaren berdina da. </p> </blockquote> <p>Beste modu batera esanik, <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>, aldetzat duen <a href="/wiki/Karratu" title="Karratu">karratuaren</a> azalera, triangeluan kontrako angeluzuzena osatzen duten bi aldeen, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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>, karratuen azaleren batura da. Hau da: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 a^{2}+b^{2}=c^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92333b53991e3ea02f5d6384bac4911ae3060a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.983ex; height:3.009ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2},}"></span></dd></dl> </blockquote> <p>ekuazio honi batzuetan <b>Pitagorasen ekuazioa</b> deitzen zaio.<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> </p><p>Teoremak K.a. 570 inguruan jaiotako <a href="/wiki/Pitagoras" title="Pitagoras">Pitagoras</a> filosofo <a href="/wiki/Greziar" title="Greziar">greziarraren</a> izena darama. Teorema metodo desberdin askoren bidez frogatu izan da —segur aski, edozein teorema matematikok izandako frogatzerik handiena—. Frogak askotarikoak dira, <a href="/wiki/Geometria" title="Geometria">geometrikoak</a> eta <a href="/wiki/Aljebra" title="Aljebra">aljebraikoak</a> barne, batzuk duela milaka urtekoak. </p><p><a href="/wiki/Euklidear_espazio" class="mw-redirect" title="Euklidear espazio">Espazio euklidearra</a> <a href="/wiki/Koordenatu_kartesiar" title="Koordenatu kartesiar">koordenatu kartesiarren sistema</a> batek irudikatzen duenean <a href="/wiki/Geometria_analitiko" title="Geometria analitiko">geometria analitikoan</a>, <a href="/wiki/Distantzia_euklidear" title="Distantzia euklidear">distantzia euklidearrak</a> erlazio pitagorikoa betetzen du: bi punturen arteko distantzia puntuen arteko koordenatu bakoitzean dagoen aldearen karratuen batura da. </p><p>Teorema hainbat modutan <a href="/w/index.php?title=Orokortze&action=edit&redlink=1" class="new" title="Orokortze (sortu gabe)">orokortu</a> daiteke: <a href="/wiki/Dimentsio" title="Dimentsio">dimentsio</a> gehiagoko espazioetara, <a href="/wiki/Geometria_ez-euklidear" title="Geometria ez-euklidear">euklidearrak ez diren espazioei</a>, triangelu zuzenak ez diren objektuei, eta triangeluak ez diren objektuei, baizik eta <i>n</i>-dimentsiodun solidoak. Pitagorasen teoremak matematikatik kanpo interesa erakarri du abstrakzio matematikoaren, mistikoaren edo botere intelektualaren sinbolo gisa; erreferentzia herrikoi ugari ditu literaturan, antzerki-lanetan, musikaletan, abestietan, zigiluetan eta karikaturetan. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=1" title="Aldatu atal hau: «Historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=1" title="Edit section's source code: Historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pitagorasen teoremak <a href="/wiki/Pitagoras" title="Pitagoras">Pitagoras</a> (<a href="/wiki/K.a._VI._mendea" title="K.a. VI. mendea">K.a. VI. mendea</a>) greziar filosofo eta matematikariaren ondoren darama izena, batez ere, haren froga <a href="/wiki/Eskola_pitagorikoa" title="Eskola pitagorikoa">eskola pitagorikoari</a> zor zaiolako. Dena dela, lehenagoko, <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamian</a> eta <a href="/wiki/Antzinako_Egipto" title="Antzinako Egipto">Antzinako Egipton</a>, bazuten zantzuaren aditzea, triangelu zuzenen aldeekin bat zetozen hirukote-balioak ezagutzen baitzituzten, eta triangelu horiei buruzko eragiketak ebazteko erabiltzen ziren, oholtxo eta papiro batzuetan adierazten den moduan<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. Hala ere, ez du luzaroan iraun inolako dokumenturik hori teorikoki erlazionatzen duenik. </p><p>Teoremaren historia lau zatitan bana daiteke: hirukote pitagorikoen ezagutza, triangelu zuzen baten aldeen arteko erlazioaren ezagutza, aldameneko angeluen arteko erlazioen ezagutza, eta sistema dedktibo batzuen barruan teoremak frogatzea. </p><p>Antzinako Egipton, K.a. 1800. urtean idatziriko papiro batean hirukote pitagorikoa erantzuntzat duen arazo bat biltzen du, nahiz eta triangeluekiko aipamenik ez dagoen. Mesopotamiako tauletan ere, K.a. 1800, <a href="/wiki/Larsa" title="Larsa">Larsatik</a> gertu hirukote pitagorikoekin lotura estua duten sarrera ugari aurkitu dira.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> K.a. XXVI. mendeko <a href="/wiki/Kefrenen_piramidea" title="Kefrenen piramidea">Kefrenen piramidea</a>, egiptoar triangelu sakratuan oinarriturik eraiki zen lehen piramidea izan zen, 3-4-5 proportzioan.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/India" title="India">Indian</a>, <a href="/w/index.php?title=Baudhayana_Shulba_Sutra&action=edit&redlink=1" class="new" title="Baudhayana Shulba Sutra (sortu gabe)">Baudhayana Shulba Sutra</a>-k, K.a. VIII. eta V. mende bitartean, hirukote pitagorikoen zerrenda eta Pitagorasen teoremaren enuntziatua ditu.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Askoz lehenago ezagutzen diren edukiekin, baina K.a. I. mendetik bizirik dauden testuetan, <a href="/wiki/Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a> textu txinatarrean (<i><a href="/wiki/Gnomon" title="Gnomon">Gnomonen</a> Klasiko Aritmetikoa eta Zeruko Bide Zirzularrak</i>) (3,4,5) triangeluarentzako Pitagorasen teorema bidezko arrazoimena dago. Txinan honi "<i>Gougu teorema"</i> deitzen diote.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Han_dinastia" title="Han dinastia">Han dinastiaren</a> garaian (K.a. 206tik K.o. 220ra), <i><a href="/wiki/Matematika-artearen_bederatzi_kapituluak" title="Matematika-artearen bederatzi kapituluak">Matematika-artearen bederatzi kapituluak</a></i>-en hirukote pitagorikoak agertzen dira, hauei dagozkien triangelu angeluzuzenen aipamenekin bat.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Teorema Euklidesen <i>Elementuak</i>en (I. Liburua, 47. Proposizioa) agertzen da, "Arotzaren teorema" izenez, eta bertan katetoen karratuen batura hipotenusaren karratua dela frogatzen du.<sup id="cite_ref-:1_9-0" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Bestalde, Pitagorikoek triangeluen parekotasunaren bitartez frogatu zutela uste da, nahiz eta ez gauden zihur parekotasunaren teoria garai hartan ezaguna al zen.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Frogak">Frogak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=2" title="Aldatu atal hau: «Frogak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=2" title="Edit section's source code: Frogak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pitagorasen_teorema_Birantolaketa_froga.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Pitagorasen_teorema_Birantolaketa_froga.gif/220px-Pitagorasen_teorema_Birantolaketa_froga.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Pitagorasen_teorema_Birantolaketa_froga.gif/330px-Pitagorasen_teorema_Birantolaketa_froga.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Pitagorasen_teorema_Birantolaketa_froga.gif/440px-Pitagorasen_teorema_Birantolaketa_froga.gif 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Birantolakuntza froga, berdinak diren lau triangelu zuzen erabiliz. Animazioan zati ilun zein argien azalera ez da inoiz aldatzen, ondorioz, <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> eta <i>c</i><sup>2</sup> balio bera dute.</figcaption></figure> <p>Pitagorasen teoremak froga ugari ditu, bakoitza bere metodoarekin. <a href="https://en.wikipedia.org/wiki/Elisha_Scott_Loomis" class="extiw" title="en:Elisha Scott Loomis">E.S. Loomis</a> matematikari estatubatuarrak adibidez, 1927. urtean, 367 froga batu zituen <i>The Pythogorean Proposition</i> liburuan.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Liburu horretan, Loomisek ebazpenak lau multzo handitan banatu zituen: <b>aljebraikoak</b>, non triangeluaren aldeak eta segmentuak erlazionatzen diren; <b>geometrikoak</b>, non azaleren konparaketa egiten den; <b>dinamikoak</b> indarra eta masaren propietateen bidez; eta <b>koaternioiak</b>, bektoreen bidez. </p> <div class="mw-heading mw-heading3"><h3 id="Froga_algebraikoak">Froga algebraikoak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=3" title="Aldatu atal hau: «Froga algebraikoak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=3" title="Edit section's source code: Froga algebraikoak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ondorengo froga <a href="/wiki/Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a>, K.a. 500-300 urteen artean idatzitako lan matematiko txinatarrean agertzen da, baina Pitagorasek obra honen berri izan ez zuelako ustea dago. </p><p>Froga honela doa. Demagun triangelu angeluzuzen bat dugula, <span class="mwe-math-element"><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> katetoak (orokortasunik galdu gabe, demagun <span class="mwe-math-element"><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\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span>) eta <span class="mwe-math-element"><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> hipotenusa dituena. </p><p>Beheko animazioan ikusten den bezala, <span class="mwe-math-element"><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> aldedun karratuan hasieran barnean zuen triangeluaz gain, alde bakoitzari <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> katetodun beste triangelu bat gehitzen badiogu, karratu txikiago bat lortuko dugu erdian. Barneko karratu honen aldeen luzera <span class="mwe-math-element"><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-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca61f9c918fe1deb227ed79d4979d70c443ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle b-a}"></span> da (gogoratu <span class="mwe-math-element"><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\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span> hipotesiz). Karratu baten azalera aldeen karratua denez, eta triangeluzuzen baten azalera katetoen biderketaren erdia, <span class="mwe-math-element"><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> aldedun karratuaren azalera lau triangeluren azaleren batura gehi erdiko karratuaren azalera da. Hau da, froga amaituz: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=4\left({\frac {a\cdot b}{2}}\right)+(b-a)^{2}=b^{2}+a^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <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>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=4\left({\frac {a\cdot b}{2}}\right)+(b-a)^{2}=b^{2}+a^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de18034d83105724c0229d70c59aab9918a573b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.566ex; height:6.176ex;" alt="{\displaystyle c^{2}=4\left({\frac {a\cdot b}{2}}\right)+(b-a)^{2}=b^{2}+a^{2}.}"></span></dd></dl> </blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pythagoras-2a.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/220px-Pythagoras-2a.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/330px-Pythagoras-2a.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/440px-Pythagoras-2a.gif 2x" data-file-width="590" data-file-height="590" /></a><figcaption>Zhoubiko frogaren animazioa.</figcaption></figure> <p>Antzeko froga bat ere bada hipotenusaren aldeak kanporantz definituriko karratua aztertu beharrean, barrurantz definituriko karratua aztertzen duena.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Hau da, <span class="mwe-math-element"><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> aldedun karratuaren kanpoko alde bakoitzean, lau denera, <span class="mwe-math-element"><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> hipotenusadun triangeluak jarriz gero, <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> luzeradun aldea duen karratua lortuko dugu. Beraz, <span class="mwe-math-element"><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)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ceb5efe73b6089f83653aacb8db72a3dcc0d49b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.931ex; height:3.176ex;" alt="{\displaystyle (a+b)^{2}}"></span> azaleraduna. Lau triangeluek eta <span class="mwe-math-element"><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> aldedun karratuak karratu haundienaren azalera bera izan behar duenez: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 (a+b)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <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>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf3d4a345c017aab39a9b3798bfd2ab804f7ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.194ex; height:5.343ex;" alt="{\displaystyle (a+b)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,}"></span></dd></dl> </blockquote> <p>Pitagorasen ekuazioa ondorioztatuz </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=(a+b)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>=</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>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=(a+b)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6155af4b2e15ce0791c4db259ee2ab09b426b983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.978ex; height:3.176ex;" alt="{\displaystyle c^{2}=(a+b)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.}"></span></dd></dl> </blockquote> <p><br /> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pythagoras_algebraic2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Pythagoras_algebraic2.svg/220px-Pythagoras_algebraic2.svg.png" decoding="async" width="220" height="566" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Pythagoras_algebraic2.svg/330px-Pythagoras_algebraic2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Pythagoras_algebraic2.svg/440px-Pythagoras_algebraic2.svg.png 2x" data-file-width="502" data-file-height="1292" /></a><figcaption>Pitagorasen teoremaren froga diagramatikoa.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Froga_geometrikoak">Froga geometrikoak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=4" title="Aldatu atal hau: «Froga geometrikoak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=4" title="Edit section's source code: Froga geometrikoak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Triangeluen_antzekotasuna">Triangeluen antzekotasuna</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=5" title="Aldatu atal hau: «Triangeluen antzekotasuna»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=5" title="Edit section's source code: Triangeluen antzekotasuna"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg/220px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg.png" decoding="async" width="220" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg/330px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg/440px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras.svg.png 2x" data-file-width="310" data-file-height="330" /></a><figcaption>Triangeluen parekotasunaren eskema.</figcaption></figure> <p>Froga hasi aurretik, gogoan izan bi triangeluren angeluak kongruenteak badira, <a href="/wiki/Antzekotasun_(geometria)" title="Antzekotasun (geometria)">antzeko triangeluak</a> direla. Kasu honetan, dagozkien aldeen luzerak proportzionalak dira. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> triangelua bi triangeluzuzenetan zatitu daiteke, <span class="mwe-math-element"><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> angeluan oinarriarekiko zuzena botaz, irudian ikusten den moduan. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></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 AHC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>H</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AHC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae16ac762889ff3721868723e8a37dd10fc6a89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.573ex; height:2.176ex;" alt="{\displaystyle AHC}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BHC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>H</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BHC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4497fa689e14ff8017cc5b8188b2c6069518ab3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.594ex; height:2.176ex;" alt="{\displaystyle BHC}"></span> triangelu angeluzuzenek angelu kongruenteak dituzte: <span class="mwe-math-element"><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> angeluzuzena, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle {AHC}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>H</mi> <mi>C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {AHC}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd77d8341fc902befe1226e35b728df076bd833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.251ex; height:2.176ex;" alt="{\displaystyle \angle {AHC}}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle {BHC}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>H</mi> <mi>C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle {BHC}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c62f235b68e4cb213fba44e9d3bf0c4dc4fc7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.272ex; height:2.176ex;" alt="{\displaystyle \angle {BHC}}"></span> angeluzuzenez ordezkatu dugu. Ondorioz, esandako triangeluak antzekoak dira, ondorengo ekuazioak betez </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AHC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>H</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AHC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae16ac762889ff3721868723e8a37dd10fc6a89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.573ex; height:2.176ex;" alt="{\displaystyle AHC}"></span>-ren arteko antzekotasuna:</li></ul> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b}{b'}}={\frac {c}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b}{b'}}={\frac {c}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c99a5852f22a6d356d8fb8e0d192ace1fbd5dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.107ex; height:5.509ex;" alt="{\displaystyle {\frac {b}{b'}}={\frac {c}{b}},}"></span></dd></dl> </blockquote> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \Rightarrow b^{2}=b'c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow b^{2}=b'c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d892e3bb7c13fc40e096dfcbe47fcfd67ff752f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.455ex; height:2.676ex;" alt="{\displaystyle \Rightarrow b^{2}=b'c.}"></span></dd></dl> </blockquote> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BHC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>H</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BHC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4497fa689e14ff8017cc5b8188b2c6069518ab3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.594ex; height:2.176ex;" alt="{\displaystyle BHC}"></span>-ren arteko antzekotasuna:</li></ul> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{a'}}={\frac {c}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{a'}}={\frac {c}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b42b0933fdaf00d409981cf3d004af8918a716b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.339ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{a'}}={\frac {c}{b}},}"></span></dd></dl> </blockquote> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \Rightarrow a^{2}=a'c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow a^{2}=a'c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed7c65c1c54b0c108caa562edc2df704bfe39ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.92ex; height:2.676ex;" alt="{\displaystyle \Rightarrow a^{2}=a'c.}"></span></dd></dl> </blockquote> <p>Aurreko bi ekuazioak batuz gero: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 a^{2}+b^{2}=a'c+b'c=c(a'+b'),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>c</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>c</mi> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=a'c+b'c=c(a'+b'),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4a7a5730cc3b194d04d4537a3751d14d93403b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.724ex; height:3.176ex;" alt="{\displaystyle a^{2}+b^{2}=a'c+b'c=c(a'+b'),}"></span></dd></dl> </blockquote> <p>baina <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <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/cb7f571796900d7d0407984060178627b43d6cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.352ex; height:3.009ex;" alt="{\displaystyle (a'+b')=c}"></span>  denez, azkenik, Pitagorasen ekuazioa lortzen da: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 a^{2}+b^{2}=c^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b56b985c78deb115014efe90ce634d73dd51fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.983ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}.}"></span></dd></dl> </blockquote> <div class="mw-heading mw-heading4"><h4 id="Einsteinen_froga_trigonometrikoa">Einsteinen froga trigonometrikoa</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=6" title="Aldatu atal hau: «Einsteinen froga trigonometrikoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=6" title="Edit section's source code: Einsteinen froga trigonometrikoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Einstein-trigonometric-proof.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Einstein-trigonometric-proof.svg/220px-Einstein-trigonometric-proof.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Einstein-trigonometric-proof.svg/330px-Einstein-trigonometric-proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Einstein-trigonometric-proof.svg/440px-Einstein-trigonometric-proof.svg.png 2x" data-file-width="387" data-file-height="312" /></a><figcaption>Hipotenusako angeluzuzenaren zatiketa, Einsteinen frogan bezala.</figcaption></figure> <p>Antzeko triangeluen frogaren moduan, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstenen</a> frogak ere hipotenusako angeluzuzena zatitzen du aurkako aldera perpendikularra sortuz. Triangelu angeluzuzen baten barruko ratioa <a href="/wiki/Sinu" title="Sinu">sinuaren</a> definizioan erabiliz gero: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 )={\frac {a}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha )={\frac {a}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3bd0a009053e94b5922cdc3041c7a1d0fa408e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.964ex; height:4.676ex;" alt="{\displaystyle \sin(\alpha )={\frac {a}{c}},}"></span></dd></dl> </blockquote> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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(\beta )={\frac {b}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\beta )={\frac {b}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4b9dd477342df376f0fabc3f534982f9aaaa75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.585ex; height:5.343ex;" alt="{\displaystyle \sin(\beta )={\frac {b}{c}},}"></span></dd></dl> </blockquote> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=b\sin(\beta )+a\sin(\alpha )={\frac {b^{2}}{c}}+{\frac {a^{2}}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=b\sin(\beta )+a\sin(\alpha )={\frac {b^{2}}{c}}+{\frac {a^{2}}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236f87bfb9e76cc133b44d000c4718ac6a26f807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.691ex; height:5.676ex;" alt="{\displaystyle c=b\sin(\beta )+a\sin(\alpha )={\frac {b^{2}}{c}}+{\frac {a^{2}}{c}},}"></span></dd></dl> </blockquote> <p>eta beraz </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=a^{2}+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</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>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10f9b628ee334c9ec8dcf1f54db092787bebdc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.983ex; height:2.843ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}.}"></span></dd></dl> </blockquote> <div class="mw-heading mw-heading4"><h4 id="Froga_grafikoa">Froga grafikoa</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=7" title="Aldatu atal hau: «Froga grafikoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=7" title="Edit section's source code: Froga grafikoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ezkerreko irudian ikusten den <span class="mwe-math-element"><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> aldedun triangelu angeluzuzen eta katetoei eta hipotenusari dagokien karratuetatik abiatuz, bi karratu ezberdin eraikitzen dira: </p> <ul><li>Horietako bat katetoen karratuez eta hasierako triangeluaren berdinak diren beste lau triangelu angeluzuzenez dago osaturik (erdiko irudia).</li></ul> <ul><li>Beste karratua aurreko lau triangeluek eta hipotenusaren karratuak osatzen dute (eskuineko irudia).</li></ul> <p>Hauetako karratu bakoitzari triangeluak kentzen badizkiogu, nabarmena da azalera griseko karratua <span class="mwe-math-element"><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^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65c32a92faefe284e22fe9026456101800425e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.87ex; height:3.176ex;" alt="{\displaystyle (c^{2})}"></span>, karratu urdin eta horiaren <span class="mwe-math-element"><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^{2}+a^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b^{2}+a^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5436a7bd5cb039dc5b9e709818a2736a448c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.986ex; height:3.176ex;" alt="{\displaystyle (b^{2}+a^{2})}"></span> baliokidea dela. Horrela, Pitagorasen teorema frogatzen da. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg/220px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg.png" decoding="async" width="220" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg/330px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg/440px-Teorema_de_Pit%C3%A1goras.Pit%C3%A1goras_b.svg.png 2x" data-file-width="310" data-file-height="150" /></a><figcaption>Zentroko karratuen baturak eta eskuinekoak azalera bera dute.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Froga_analitikoa">Froga analitikoa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=8" title="Aldatu atal hau: «Froga analitikoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=8" title="Edit section's source code: Froga analitikoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Kalkulu_diferentzial" title="Kalkulu diferentzial">Kalkulu diferentziala</a> erabiliz ere lortu dezakegu teoremaren frogarik. Kalkuluko notazioarekin bat egiteko, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> triangeluaren aldeei izena aldatuko diegu, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y:=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>:=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y:=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e2577f93495f7a7e6e017a95d9466a02da6872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.908ex; height:2.009ex;" alt="{\displaystyle y:=c}"></span> deituz hipotenusaren luzerari eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e808db52c31ff788c405fb855eeede04c5eace94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.073ex; height:2.176ex;" alt="{\displaystyle x:=b}"></span> kateto bertikalari diagraman ikusi daitekeen moduan. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b930d133ca536a071bec52a9acc4b05482890d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle AC}"></span> aldea apur bat luzatuz gero, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b637d7a6d64d797391d40ceae9e8696e5d76f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x}"></span> kopuruz, orduan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> ere apur bat haundituko da, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ADB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>D</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ADB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a597174e1a2976908c8c401664a5ad49d226e9fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.431ex; height:2.176ex;" alt="{\displaystyle ADB}"></span> triangelua sortuz. <span class="mwe-math-element"><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> puntutik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3357f928ef7b5ffe20a313be8150b6bcbe084bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.54ex; height:2.176ex;" alt="{\displaystyle BE}"></span> aldearekiko perpendikularra hartuz gero, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CDE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CDE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08701a099cea4d37f1e15ea945234e2c1ef25c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.466ex; height:2.176ex;" alt="{\displaystyle CDE}"></span> triangelua sortua dugu, gutxi gorabehera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> triangeluaren antzekoa. Beraz, triangeluen antzekotasuna erabiliz, aldeen proportzioa berdina izan beharko luke, hau da: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {x}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {x}{y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567e627b63d1a330f24d30b54b878fee0135d987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.369ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {x}{y}}.}"></span></dd></dl> </blockquote> <p>Honek <a href="/wiki/Ekuazio_diferentzial" title="Ekuazio diferentzial">ekuazio diferentzial</a> batera garamatza, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\mathrm {d} y=x\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\mathrm {d} y=x\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e7c0b76340ba213429d72a68e0b57ac35019db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.654ex; height:2.509ex;" alt="{\displaystyle y\mathrm {d} y=x\mathrm {d} x}"></span>, zuzenean ebatzi dezakeguna integratuz: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \int y\mathrm {d} y=\int x\mathrm {d} x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mo>∫</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int y\mathrm {d} y=\int x\mathrm {d} x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d4f442bc4aea20f8dd64735563964bcb4534fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.462ex; height:5.676ex;" alt="{\displaystyle \int y\mathrm {d} y=\int x\mathrm {d} x,}"></span></dd></dl> </blockquote> <p>ondorioz </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 y^{2}=x^{2}+C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{2}+C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87aebfb29f8e93639573592f08a118a73e5c3f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.951ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{2}+C.}"></span></dd></dl> </blockquote> <p>Konstantea erraz aurkitu dezakegu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8316c9c21e9993c1d7e2cf421cee892743b4d07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.484ex; height:2.009ex;" alt="{\displaystyle y=a}"></span> hartuz, hau da, <span class="mwe-math-element"><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=a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df092aa7114e217520e825f07c5bd3e224e7b928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.676ex;" alt="{\displaystyle C=a^{2}}"></span>. Froga hau intuitiboa da, baina zehatz egin daiteke limiteak hartuz, erreferentzian ikus daitekeen moduan.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pythag_differential_proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pythag_differential_proof.svg/220px-Pythag_differential_proof.svg.png" decoding="async" width="220" height="323" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pythag_differential_proof.svg/330px-Pythag_differential_proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pythag_differential_proof.svg/440px-Pythag_differential_proof.svg.png 2x" data-file-width="483" data-file-height="709" /></a><figcaption>Pitagorasen teoremaren froga diferentziala.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Teoremaren_alderantzizkoa">Teoremaren alderantzizkoa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=9" title="Aldatu atal hau: «Teoremaren alderantzizkoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=9" title="Edit section's source code: Teoremaren alderantzizkoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pitagorasen teoremaren alderantzizkoa ere egia da. Hau da, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> luzerako aldeak dituen triangelu bat emanik, <span class="mwe-math-element"><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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span> betetzen bada, orduan <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> aldeen arteko angelua angelu zuzena da. </p><p>Teorema hau Euklidesen <i>Elementuak</i>en (I. Liburua, 48. Proposizioa)<sup id="cite_ref-:1_9-1" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> ageri da, aldeak definitzen duten karratuei erreferentzia eginez: "Triangelu baten aldeetako bateko karratua triangeluaren gainerako bi aldeetako karratuen batura berdina bada, orduan gainerako bi aldeek duten angelua angelu zuzena da." </p><p>Proba kosinuaren teorema erabiliz froga daiteke. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> triangelu orokor bat emanik, non <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> aldeek pitagorasen erlazioa betetzen duten <span class="mwe-math-element"><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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span>, baina triangeluak ez duelarik nahitaez angelu zuzenik. Bestalde, beti sor dezakegu beste triangelu <span class="mwe-math-element"><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"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </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/dd38dba73f9da63e1c7709de40805d82cd55c255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle A'B'C'}"></span> bat, <span class="mwe-math-element"><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'=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237efff16ba3387b185337e428e9d422ac3943a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.243ex; height:2.509ex;" alt="{\displaystyle a'=a}"></span> eta <span class="mwe-math-element"><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'=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b'=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbb42e4a3582f02d86265b686aa6cd9ab1a5b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.778ex; height:2.509ex;" alt="{\displaystyle b'=b}"></span> aldeak dituena, angelu zuzen bat izanik alde honetan, eta orain pitagorasen teoremaren ondorioz triangelu honen hipotenusak </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'={\sqrt {a^{2}+b^{2}}}=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c'={\sqrt {a^{2}+b^{2}}}=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0eab7ea163be77387cd458d65332b479af6364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.395ex; height:3.509ex;" alt="{\displaystyle c'={\sqrt {a^{2}+b^{2}}}=c}"></span></dd></dl> </blockquote> <p>beteko du. Beraz alde berdinak dituzten bi triangelu ditugu, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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>, <a href="/wiki/Kosinuaren_teorema" title="Kosinuaren teorema">kosinuaren teoremaren</a> ondorioz, alegia triangeluaren angelu bakoitza hiru aldeek bakarrik zehazten dutenez, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> triangeluko <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> aldeen arteko angelua <span class="mwe-math-element"><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"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </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/dd38dba73f9da63e1c7709de40805d82cd55c255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle A'B'C'}"></span> triangeluaren berdina da, hau da, angelu zuzena. </p> <div class="mw-heading mw-heading2"><h2 id="Erabilera_adibideak">Erabilera adibideak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=10" title="Aldatu atal hau: «Erabilera adibideak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=10" title="Edit section's source code: Erabilera adibideak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Hirukote_pitagorikoak">Hirukote pitagorikoak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=11" title="Aldatu atal hau: «Hirukote pitagorikoak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=11" title="Edit section's source code: Hirukote pitagorikoak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Hirukote pitagorikoak</b> hiru <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">zenbaki oso</a> positiboz <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> osatua dago, non pitagorasen erlazioa betetzen duten <span class="mwe-math-element"><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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span>. Horrelako hirukotea <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}"></span> idatzi ohi da, adibide ezagun bat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,4,5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,4,5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfdcfb9811db516be2f4628b2171a7721a6edf84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (3,4,5)}"></span> izanik. </p><p>Gainera, hiru zenbakiak <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> <a href="/wiki/Zenbaki_elkarrekiko_lehenak" title="Zenbaki elkarrekiko lehenak">elkarrekiko lehenak</a> badira (hau da, <a href="/wiki/Zatitzaile_komun_handiena" title="Zatitzaile komun handiena">zatitzaile komun handiena</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> da), <b>hirukote pitagoriko primitiboa</b> deitzen diogu. Adibidez, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,4,5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,4,5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfdcfb9811db516be2f4628b2171a7721a6edf84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (3,4,5)}"></span> hirukote pitagoriko primitibo bat da, baina <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (6,8,10)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (6,8,10)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f701ba12069733c44a64d51a8803fc3bcab27854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle (6,8,10)}"></span> ez. </p><p>Hirukote pitagorikoek triangelu zuzen baten hiru alde osoen luzera, hau da, aldearen luzera zenbaki oso bat da, deskribatzen dute. Hala ere, alde osoak ez diren triangelu zuzenek ez dute hirukote pitagorikoa osatzen. Adibidez, <span class="mwe-math-element"><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=1=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b29e6a2c379a5ddc2a15638fc8611f5f76e8d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.587ex; height:2.176ex;" alt="{\displaystyle a=1=b}"></span> eta <span class="mwe-math-element"><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={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5e1062eb27627bc4f0c116ab3113da49e1636f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.204ex; height:3.009ex;" alt="{\displaystyle c={\sqrt {2}}}"></span> aldeak dituen triangelua angeluzuzena da, baina ez da hirukote pitagorikoa, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> ez baita zenbaki oso bat. </p><p>Soluzio osoak bilatzean, <span class="mwe-math-element"><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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span> <a href="/wiki/Ekuazio_diofantoar" title="Ekuazio diofantoar">ekuazio diofantoarra</a> da. Beraz, hirukot epitagorikoak ekuazio diofantino ez-lineal baten soluzio zaharrenen artean daude. </p> <div class="mw-heading mw-heading3"><h3 id="Zenbaki_neurtezinak_eraikitzea">Zenbaki neurtezinak eraikitzea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=12" title="Aldatu atal hau: «Zenbaki neurtezinak eraikitzea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=12" title="Edit section's source code: Zenbaki neurtezinak eraikitzea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Zenbaki_irrazional" title="Zenbaki irrazional">Zenbaki irrazional</a>»</div> <p>Pitagorasen teoremaren ondorioetako klasikoetako bat, <a href="/w/index.php?title=Zenbaki_neurtezin&action=edit&redlink=1" class="new" title="Zenbaki neurtezin (sortu gabe)">zenbaki neurtezinak</a> (<a href="/wiki/Zenbaki_arrazional" title="Zenbaki arrazional">arrazionalak</a> ez diren zenbaki <a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">errealak</a>) zuzen eta konpasa erabiliz eraiki daitezkeela da. Honek <a href="/wiki/Eskola_pitagorikoa" title="Eskola pitagorikoa">eskola Pitagorikoaren</a> zenbakien kontzueptuarekin talka egiten zuen, hauek proportzioak zenbaki osoen zatiketaz aztertzen baitzituzten (hau da, zenbaki arrazionalak bakarrik onartzen zituzten).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Kondairak dio, pitagorikoek <a href="/wiki/Hipaso" title="Hipaso">Hipaso</a> (K.a. 500) itsasoan itoarazi zutela, zenbaki irrazionalen existentzia ezagutarazteagatik.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Zehazki, luzera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> duten bi alde izanez gero, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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>, zenbaki neurtezinak Pitagorasen teorema erabiliz eraiki daitezke, hipotenusa zenbaki irrazional neurtezina baita, <span class="mwe-math-element"><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={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5e1062eb27627bc4f0c116ab3113da49e1636f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.204ex; height:3.009ex;" alt="{\displaystyle c={\sqrt {2}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Zenbaki_konplexuak">Zenbaki konplexuak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=13" title="Aldatu atal hau: «Zenbaki konplexuak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=13" title="Edit section's source code: Zenbaki konplexuak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Zenbaki_konplexu" title="Zenbaki konplexu">Zenbaki konplexu</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Complex_conjugate_picture.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/220px-Complex_conjugate_picture.svg.png" decoding="async" width="220" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/330px-Complex_conjugate_picture.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/440px-Complex_conjugate_picture.svg.png 2x" data-file-width="300" data-file-height="422" /></a><figcaption>Zenbaki konplexu baten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> moduluaren <span class="mwe-math-element"><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> adierazpen grafikoa.</figcaption></figure> <p>Jakina da zenbaki konplexu baten </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 z=x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}"></span>,</dd></dl> </blockquote> <p><a href="/wiki/Balio_absolutu" title="Balio absolutu">balio absolutua</a>, zenbaki konplexuen kasuan modulua ere deitua, parte erreal eta irudikariaren karratuen baturaren erroa dela </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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:=|z|={\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:=|z|={\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf49fad45ed3f2a9b326439ee6499d9caca37c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.037ex; height:4.843ex;" alt="{\displaystyle r:=|z|={\sqrt {x^{2}+y^{2}}}}"></span>.</dd></dl> </blockquote> <p>Beste era batera esanik, zenbaki konplexuen moduluak, parte errealak eta irudikariak hirukote pitagorikoa osatzen dute. Hau da, </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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^{2}=x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}=x^{2}+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb2a5f9d5d6102af609d3730471b017b2ac7ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle r^{2}=x^{2}+y^{2}}"></span>.</dd></dl> </blockquote> <p>Bada <span class="mwe-math-element"><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> geometrikoki, ardatz horizontala zati erreal bezala eta bertikala irudikari bezala, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> puntuak onarriarekiko <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> luzera errepresentatzen du. </p> <div class="mw-heading mw-heading3"><h3 id="Geometria_euklidearra">Geometria euklidearra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=14" title="Aldatu atal hau: «Geometria euklidearra»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=14" title="Edit section's source code: Geometria euklidearra"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Koordenatu_kartesiar" title="Koordenatu kartesiar">Koordenatu kartesiarretan</a> ezarririko bi punturen arteko distantziaren ekuazioa Pitagorasen teoremaren ondorioa da. Plano errealeko bi punturen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{x})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7960934ab458aade7a2d1af20d8ca39137774fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.539ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{x})}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> arteko distantzia, batzuetan distantzia Euklidearra deitua, honakoa da </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |(x_{1},y_{1})-(x_{2},y_{2})|={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |(x_{1},y_{1})-(x_{2},y_{2})|={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4539bc3556dc40275353112ae658b6de2dd2a089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:47.801ex; height:4.843ex;" alt="{\displaystyle |(x_{1},y_{1})-(x_{2},y_{2})|={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}}"></span>.</dd></dl> </blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Distantzia_euklidearra.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Distantzia_euklidearra.png/220px-Distantzia_euklidearra.png" decoding="async" width="220" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Distantzia_euklidearra.png/330px-Distantzia_euklidearra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Distantzia_euklidearra.png/440px-Distantzia_euklidearra.png 2x" data-file-width="451" data-file-height="389" /></a><figcaption>Koordenatu kartesiarretan bi punturen arteko distantzia euklidearraren azalpen geometrikoa.</figcaption></figure> <p>Irudian ikus daitekeen bezala, bi punturen distantzia hauek osatzen duten triangelu angeluzuzenaren hipotenusatzat uler daiteke. Hala interpretatuz gero, triangeluaren bi katetoek <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{1}-x_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{1}-x_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2386becdaa27a7557ccc33e04a518b4f99636d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle |x_{1}-x_{2}|}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y_{1}-y_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y_{1}-y_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b9575cd658a4ff8b5c6af77d895fbeddb1c5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle |y_{1}-y_{2}|}"></span> luzera izango dute hurrenez hurrun. Balio absolutuaren karratuak sinua errespetatzen duenez, </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f16f76f572b10e3e010bde1b7985bff63566b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.209ex; height:3.176ex;" alt="{\displaystyle c^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.}"></span></dd></dl> </blockquote> <p>Distantziaren nozio hau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimentsiodun espazio Euklidearretara orokortu daiteke. Bertan puntuak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> dimentsiodun bektoreak dira, adibidez <span class="mwe-math-element"><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=(a_{1},\ldots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a_{1},\ldots ,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e95627e29727d4d35fa75023197fc91ebf19493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.561ex; height:2.843ex;" alt="{\displaystyle A=(a_{1},\ldots ,a_{n})}"></span> eta <mathB=(b_1,\ldots,b_n)</math>. Pitagorasen teoremaren orokortzean, bi bektore hauen arteko distantzia honako ekuazioaz definitzen da: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{1})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4482eb507a768b1b2e5d24568cfebc9a98f1de1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:63.225ex; height:7.509ex;" alt="{\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{1})^{2}}}.}"></span></dd></dl> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Pitagorasen_identitate_trigonometrikoa">Pitagorasen identitate trigonometrikoa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=15" title="Aldatu atal hau: «Pitagorasen identitate trigonometrikoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=15" title="Edit section's source code: Pitagorasen identitate trigonometrikoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diagraman ikusi daitekeen moduan, triangelu angeluzuzen batean <span class="mwe-math-element"><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> katekoaren eta <span class="mwe-math-element"><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> hipotenusaren arteko <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> angeluaren sinua eta kosinua aldeen ratioak definitzen ditu </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \theta ={\frac {b}{c}},\quad \cos \theta ={\frac {a}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta ={\frac {b}{c}},\quad \cos \theta ={\frac {a}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d646102b57a24754318119d66cf9a300e5c0699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.031ex; height:5.343ex;" alt="{\displaystyle \sin \theta ={\frac {b}{c}},\quad \cos \theta ={\frac {a}{c}}.}"></span></dd></dl> </blockquote> <p>Identitate trigonometrikoa beraz Pitagorasen teoremaren ondorioa da<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> zeren </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 ^{2}\theta +\cos ^{2}\theta ={\frac {a^{2}}{c^{2}}}+{\frac {b^{2}}{c^{2}}}=1.}"> <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>θ</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta ={\frac {a^{2}}{c^{2}}}+{\frac {b^{2}}{c^{2}}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca433c6cb1c50573ca4d79892f7ee2533f40f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.735ex; height:6.009ex;" alt="{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta ={\frac {a^{2}}{c^{2}}}+{\frac {b^{2}}{c^{2}}}=1.}"></span></dd></dl> </blockquote> <p>Bestalde, antzeko triangeluetan aldeen arteko proportzioek angeluekiko mendekotasuna dute, triangeluen azalera edozein dela ere. Beraz hipotenusaren luzera unitatea duen triangelu angeluzuzen baten katetoek <span class="mwe-math-element"><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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }"></span> eta <span class="mwe-math-element"><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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"></span> luzera dute, irudian ikusi daitekeen moduan. </p> <div class="mw-heading mw-heading3"><h3 id="Biderketa_bektoriala">Biderketa bektoriala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=16" title="Aldatu atal hau: «Biderketa bektoriala»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=16" title="Edit section's source code: Biderketa bektoriala"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Cross_product_parallelogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cross_product_parallelogram.svg/220px-Cross_product_parallelogram.svg.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cross_product_parallelogram.svg/330px-Cross_product_parallelogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cross_product_parallelogram.svg/440px-Cross_product_parallelogram.svg.png 2x" data-file-width="794" data-file-height="593" /></a><figcaption>Biderketa eskalarra eta bektorialaren arteko erlazioa.</figcaption></figure> <p>Pitagorasen teoremak <a href="/wiki/Biderketa_eskalar" title="Biderketa eskalar">biderketa eskalarra</a> eta <a href="/wiki/Biderketa_bektorial" title="Biderketa bektorial">biderketa bektorialaren</a> arteko erlazioa dakar: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7f4a7fc23bb2377a327090ead0a8b7619a9673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.781ex; height:3.176ex;" alt="{\displaystyle \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},}"></span></dd></dl> </blockquote> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> bektoreak direlarik. Gogoratu operazio bilinear hauen definizioak honakoak direla </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} &=ab\mathbf {n} \sin {\theta }\\\mathbf {a} \cdot \mathbf {b} &=ab\cos {\theta },\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} &=ab\mathbf {n} \sin {\theta }\\\mathbf {a} \cdot \mathbf {b} &=ab\cos {\theta },\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae084d2d8b72996221da029264e391a601d061a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.908ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} &=ab\mathbf {n} \sin {\theta }\\\mathbf {a} \cdot \mathbf {b} &=ab\cos {\theta },\end{aligned}}}"></span></dd></dl> </blockquote> <p>non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> bektore unitarioa, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> bektoreekiko normala den. Erlazioa beraz Pitagorasen teorema eta identitate trigonometrikoaren ondorioa da </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <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 \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=a^{2}b^{2}\mathbf {n} ^{2}\sin ^{2}{\theta }+a^{2}b^{2}\cos ^{2}{\theta }=a^{2}b^{2}=\|a\|^{2}\|b\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>a</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">‖</mo> <mi>b</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=a^{2}b^{2}\mathbf {n} ^{2}\sin ^{2}{\theta }+a^{2}b^{2}\cos ^{2}{\theta }=a^{2}b^{2}=\|a\|^{2}\|b\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e43c733a1f96f92aff282448deec091fe1557fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.613ex; height:3.176ex;" alt="{\displaystyle \|{\textbf {a}}\times {\textbf {b}}\|^{2}+(\mathbf {a} {\dot {\mathbf {b} }})^{2}=a^{2}b^{2}\mathbf {n} ^{2}\sin ^{2}{\theta }+a^{2}b^{2}\cos ^{2}{\theta }=a^{2}b^{2}=\|a\|^{2}\|b\|^{2}.}"></span></dd></dl> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Pitagorasen_alderantzizko_teorema">Pitagorasen alderantzizko teorema</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=17" title="Aldatu atal hau: «Pitagorasen alderantzizko teorema»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=17" title="Edit section's source code: Pitagorasen alderantzizko teorema"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Triangelu angeluzuzen batean, pitagorasen alderantzizko ekuazioak bi katetoak, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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>, eta altuera <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> erlazionatzen ditu<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{d^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{d^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03bf521052d93d1e308e70ce9c0f1445fe4539af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.702ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{d^{2}}}.}"></span></dd></dl> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Gehiago">Gehiago</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=18" title="Aldatu atal hau: «Gehiago»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=18" title="Edit section's source code: Gehiago"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Eskailera baten neurria kalkulatzeko; eskuratu nahi den hormaren h altuera eta erpinetik (lurzoru-horma) eskaileraren oinera dagoen p distantziak ezagutzen dira.</li></ul> <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 e^{2}=h^{2}+p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}=h^{2}+p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27867da45fb51b356981fb9c567f2cb8a7048998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.693ex; height:3.009ex;" alt="{\displaystyle e^{2}=h^{2}+p^{2}}"></span></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 e={\sqrt {h^{2}+p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {h^{2}+p^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10b288ceab96821c5b6b1d3dbed73fa616d71eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.963ex; height:4.843ex;" alt="{\displaystyle e={\sqrt {h^{2}+p^{2}}}}"></span></dd></dl> <ul><li>Geometria analitiko lauan,  <span class="mwe-math-element"><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(x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f558d70cab868ac335bdf48088675900e75865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.187ex; height:2.843ex;" alt="{\displaystyle C(x_{1},y_{1})}"></span>eta <span class="mwe-math-element"><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(x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47585fa2ce9e876ce706e697f23a7b71c166e3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.345ex; height:2.843ex;" alt="{\displaystyle D(x_{2},y_{2})}"></span>puntuen artean distantzia aurkitzeko.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li></ul> <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 {CD}^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {CD}^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e547938cb35f5753eb055307ebf488d45888216f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.246ex; height:3.176ex;" alt="{\displaystyle {CD}^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}"></span></dd></dl> <ul><li>Geometrian, triangelu aldekide baten altuera aldearen menpe kalkulatzeko; ertza erabiliz tetraedro erregular baten altuera lortzeko. Zirkunskribatutako zirkunferentziaren erradioa ezagututa, inskribatutako triangelu aldekide eta hexagono erregular baten apotema aurkitzeko.</li> <li>Aljebran, zenbaki oso gaussiar bat lehena den aztertzeko. Adibidez, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1+4i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1+4i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982d6622665da633450c31ddb2b3fd7bf2083887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.554ex; height:2.343ex;" alt="{\displaystyle \alpha =1+4i}"></span>, haren norma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\alpha )=1^{2}+4^{2}=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\alpha )=1^{2}+4^{2}=17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a96060f8110e7dc44ed4b93f9ac1b410defc8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.156ex; height:3.176ex;" alt="{\displaystyle N(\alpha )=1^{2}+4^{2}=17}"></span> da.</li> <li>Arkeologoek Pitagorasen Teorema erabiltzen dute indusketetan. Indusketa bat hasten dutenean, sareta laukizuzen bat jartzen dute induskatu beharreko azaleraren gainean. Sareta zehatz bat izateko, oinarrizko lerroen luzera erabaki ondoren (eje-X eta eje-Y), diagonalaren luzera Pitagorasen Teorema erabiliz kalkulatzen da<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>, koadrantea laukizuzena dela eta ez beste paralelogramo bat ziurtatzeko. Gainjarritako sareta <a href="/wiki/Kartesiar_koordenatu" class="mw-redirect" title="Kartesiar koordenatu">Koordenatu Kartesiarren</a> sistema gisa erabiltzen dute.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Orokortzeak">Orokortzeak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=19" title="Aldatu atal hau: «Orokortzeak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=19" title="Edit section's source code: Orokortzeak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Kosinuaren_teorema">Kosinuaren teorema</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=20" title="Aldatu atal hau: «Kosinuaren teorema»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=20" title="Edit section's source code: Kosinuaren teorema"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Kosinuaren_teorema" title="Kosinuaren teorema">Kosinuaren teorema</a>»</div> <p>Pitagorasen teorema, edozein triangelutan aldeen luzerak erlazionatzen dituen teorema orokorragoaren kasu berezi bat da, <a href="/wiki/Kosinuaren_teorema" title="Kosinuaren teorema">kosinuaren teorema</a>. Honakoa dio, <span class="mwe-math-element"><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> eta <span class="mwe-math-element"><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> aldeak dituen triangelu batean </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><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^{2}+b^{2}-2ab\cos \theta =c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}-2ab\cos \theta =c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efbf416be63eefbb9a07032cc5c480d93d9d663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.542ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}-2ab\cos \theta =c^{2}}"></span>, </p> </blockquote> <p>non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></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 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> eta <span class="mwe-math-element"><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> aldeen arteko angelua den. Alde hauek ortogonalak direnean, hau da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4628b0f731f81bfabcd8edaca00aa186f03bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \theta =\pi /2}"></span> eta ondorioz <span class="mwe-math-element"><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 \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee56ea9827922bb74fb500f33068ddb58273ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.85ex; height:2.176ex;" alt="{\displaystyle \cos \theta =0}"></span>, kosinuaren teorema Pitagorasen teoremara murrizten da. </p> <div class="mw-heading mw-heading3"><h3 id="Barne_produktudun_espazioak">Barne produktudun espazioak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=21" title="Aldatu atal hau: «Barne produktudun espazioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=21" title="Edit section's source code: Barne produktudun espazioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pitagorasen teorema <a href="/w/index.php?title=Barne_produktu&action=edit&redlink=1" class="new" title="Barne produktu (sortu gabe)">barne produktudun</a> espazioetara orokortu daiteke, berean barne produktudun espazio ezagunenak <a href="/wiki/Hilberten_espazio" title="Hilberten espazio">Hilberten espazioak</a> izanik, <a href="/wiki/Euklidear_espazio" class="mw-redirect" title="Euklidear espazio">espazio euklidearren</a> orokortzeak. Adibidez, funtzio bat barne produktudun espazio batean infinitu osagai dituen <a href="/wiki/Bektore" class="mw-redirect mw-disambig" title="Bektore">bektoretzat</a> har daiteke, hau da, <a href="/wiki/Segida" title="Segida">segidatzat</a>. </p><p>Barne produktudun espazioetan aldagai perpendikularren ordez, aldagai <a href="/wiki/Ortogonal" title="Ortogonal">ortogonalez</a> hitz egiten dugu: bi bektore <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> ortogonalak diogu, haien barne produktua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v,w\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v,w\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced6f40b6f52b3d974786df295649b5c00e51708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.635ex; height:2.843ex;" alt="{\displaystyle \langle v,w\rangle }"></span> zero baldin bada. Barne produktua bektoreen <a href="/wiki/Biderketa_eskalar" title="Biderketa eskalar"> biderketa eskalarraren</a> orokotzea da, hau izanik espazio Euklidearren barne produktu kanonikoa, baina beste batzuk ere defini ditzakegu.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Aldeen luzeraren ordez, barne produktudun espazioetan <a href="/wiki/Norma" title="Norma">norma</a> defini daiteke </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><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\|:={\sqrt {\langle v,v\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <mo fence="false" stretchy="false">‖</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|v\|:={\sqrt {\langle v,v\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c0546d498bdb2f1e6085a77b29a12d2967bafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.62ex; height:4.843ex;" alt="{\displaystyle \|v\|:={\sqrt {\langle v,v\rangle }}}"></span>. </p> </blockquote> <p>Barne produktudun espazioetan, <b>Pitagorasen teoremak</b> bi elementu ortogonalen <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span>, non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v,w\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v,w\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f9c32a68b62fe9b9422f95ce1b8bbfc6e2b0b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.896ex; height:2.843ex;" alt="{\displaystyle \langle v,w\rangle =0}"></span> (katetoak liratekenak) baturaren <span class="mwe-math-element"><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+w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v+w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b54eebfe21e642e25499aabb366f90701f9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.632ex; height:2.176ex;" alt="{\displaystyle v+w}"></span> (hipotenusa) karratua, bektore bakoitzaren luzeraren karratuen gehiketa dela dio: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><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+w\|^{2}=\|v\|^{2}+\|w\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>w</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|v+w\|^{2}=\|v\|^{2}+\|w\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3430252d4da19116ba31635f33def8b10a8cbd84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.5ex; height:3.176ex;" alt="{\displaystyle \|v+w\|^{2}=\|v\|^{2}+\|w\|^{2}}"></span>. </p> </blockquote> <p>Gainera, Pitagorasen ekuazioa bi bektore ortogonal baino gehiagoren baturara heda daiteke. Hau da, <span class="mwe-math-element"><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_{1},\dots ,v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},\dots ,v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6ddc943290a5e10aa10a064fb4d6a745f0fde7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.706ex; height:2.009ex;" alt="{\displaystyle v_{1},\dots ,v_{n}}"></span> binakako bektore ortogonalak badira (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{i},v_{j}\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{i},v_{j}\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d1ed9fcffc08273492346ff7c186a44c47c1dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.069ex; height:3.009ex;" alt="{\displaystyle \langle v_{i},v_{j}\rangle =0}"></span> edozein <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\not =j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\not =j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e183b251354fde28bb4a4b055e31499ce83f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\not =j}"></span>), Pitagorasen teoremak ondorengoa dio: </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\sum _{i=1}^{n}v_{i}\|^{2}=\sum _{i=1}^{n}\|v_{i}\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\sum _{i=1}^{n}v_{i}\|^{2}=\sum _{i=1}^{n}\|v_{i}\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26b0c066937826d4efa38c8cc5873cf52eb1937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.583ex; height:6.843ex;" alt="{\displaystyle \|\sum _{i=1}^{n}v_{i}\|^{2}=\sum _{i=1}^{n}\|v_{i}\|^{2}}"></span>. </p> </blockquote> <p>Bektore ez ortogonalentzat ere orokortu daiteke Pitagorasen teorema, norma batek <a href="/w/index.php?title=Paralelogramoaren_legea&action=edit&redlink=1" class="new" title="Paralelogramoaren legea (sortu gabe)">paralelogramoaren legea</a> betetzen diogu edozein bi bektorek ondorengo ekuazioa betetzen badute </p> <blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\|v\|^{2}+2\|w\|^{2}=\|v+w\|^{2}+\|v-w\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo fence="false" stretchy="false">‖</mo> <mi>w</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <mo>−</mo> <mi>w</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\|v\|^{2}+2\|w\|^{2}=\|v+w\|^{2}+\|v-w\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5860a8f810d230425ad1f1fd04af23e4d09e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.677ex; height:3.176ex;" alt="{\displaystyle 2\|v\|^{2}+2\|w\|^{2}=\|v+w\|^{2}+\|v-w\|^{2}}"></span>. </p> </blockquote> <p>Ohartu propietate hau bektore espazio normatuaren propietate bat dela. Hau da, paralelogramoaren legea ez dute bektore espazio normatu guztiek betetzen. Gainera, analisi funtzionaleko teorema ezaguna da espazio bektore normatu batek paralelogramoaren legea betetzen badu, norma barne produktu baten eratorria dela.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ariketak"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Fitxategi:Jakindun_logoa.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/24px-Jakindun_logoa.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/36px-Jakindun_logoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/48px-Jakindun_logoa.png 2x" data-file-width="225" data-file-height="225" /></a><figcaption></figcaption></figure> Ariketak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=22" title="Aldatu atal hau: «Ariketak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=22" title="Edit section's source code: Ariketak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-packed" style="background-color: #fef6e7; margin-left: 0;"> <li class="gallerycaption">Pitagorasen teorema</li> <li class="gallerybox" style="width: 215.33333333333px"> <div class="thumb" style="width: 213.33333333333px;"><span typeof="mw:File"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/320px--Pitagoras_teorema_azalpena_ariketaren_bidez.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="214" height="120" data-durationhint="156" data-mwtitle="Pitagoras_teorema_azalpena_ariketaren_bidez.webm" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm" type="video/webm; codecs="vp9, opus"" data-width="3840" data-height="2160" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4d/Pitagoras_teorema_azalpena_ariketaren_bidez.webm/Pitagoras_teorema_azalpena_ariketaren_bidez.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span></span></div> <div class="gallerytext"><b>Pitagorasen teorema</b> azalpena ariketaren bidez.</div> </li> <li class="gallerybox" style="width: 215.33333333333px"> <div class="thumb" style="width: 213.33333333333px;"><span typeof="mw:File"><span><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/320px--Pitagorasen_Teorema_ariketa_azalpenarekin.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="214" height="120" data-durationhint="96" data-mwtitle="Pitagorasen_Teorema_ariketa_azalpenarekin.webm" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm" type="video/webm; codecs="vp9, opus"" data-width="3840" data-height="2160" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a3/Pitagorasen_Teorema_ariketa_azalpenarekin.webm/Pitagorasen_Teorema_ariketa_azalpenarekin.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span></span></div> <div class="gallerytext"><b>Pitagorasen teorema</b> ariketa azalpenaren bidez.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=23" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=23" title="Edit section's source code: Erreferentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="http://www1.euskadi.net/harluxet/hiztegia1.asp?sarrera=Pitagorasenteorema">«Pitagorasen teorema - Harluxet Hiztegi Entziklopedikoa»</a> <i>www1.euskadi.net</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2023-06-08)</small></span></span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation" id="CITEREFSallySally2007">Sally, Judith D.; 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Elementuak. </i></a> Elhuyar <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/84-95338-26-7" title="Berezi:BookSources/84-95338-26-7">84-95338-26-7</a>.</span>.</span> </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFMaor2007">Maor, Eli.  (2007). <a rel="nofollow" class="external text" href="https://books.google.nl/books?id=Z5VoBGy3AoAC&pg=PA25&redir_esc=y#v=onepage&q&f=false"><i>The Pythagorean Theorem: A 4,000-year History. </i></a> Princeton University Press, 25 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-691-12526-8" title="Berezi:BookSources/978-0-691-12526-8">978-0-691-12526-8</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2023-06-30)</small></span></span>.</span> </li> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFLoomis1968">Loomis, Elisha Scott.  (1968). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V_o-AAAAIAAJ&newbks=0&hl=eu"><i>The Pythagorean Proposition: Its Demonstrations Analyzed and Classified, and Bibliography of Sources for Data of the Four Kinds of Proofs. </i></a> National Council of Teachers of Mathematics <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-87353-036-1" title="Berezi:BookSources/978-0-87353-036-1">978-0-87353-036-1</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2023-06-30)</small></span></span>.</span> </li> <li id="cite_note-12"><a href="#cite_ref-12">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/pythagoras/index.shtml#3">«Pythagorean Theorem and its many proofs»</a> <i>www.cut-the-knot.org</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2023-07-13)</small></span></span>.</span> </li> <li id="cite_note-13"><a href="#cite_ref-13">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFStaring1996">Staring, Mike.  (1996). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/epdf/10.1080/0025570X.1996.11996380?needAccess=true"><i>The Pythagorean Proposition: A Proof by Means of Calculus. </i></a> Mathematics Magazine, 45-46 or.</span>.</span> </li> <li id="cite_note-14"><a href="#cite_ref-14">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFShaughan1994">Shaughan, Lavine.  (1994). <a rel="nofollow" class="external text" href="https://books.google.nl/books?id=GvGqRYifGpMC&pg=PA13&redir_esc=y#v=onepage&q&f=false"><i>Understanding the infinite. </i></a> Harvard University Press, 13 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/0%3D674-92096-1" title="Berezi:BookSources/0=674-92096-1">0=674-92096-1</a>.</span>.</span> </li> <li id="cite_note-15"><a href="#cite_ref-15">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFKurt1945">Kurt, Von Fritz.  (1945). «The Discovery of Incommensurability by Hippasus of Metapontum» <i>Annals of Mathematics. Second Series</i> 46 (2): 242-264.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F1969021">10.2307/1969021</a></span>.</span>.</span> </li> <li id="cite_note-16"><a href="#cite_ref-16">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="http://www.xtec.cat/~jlagares/mates/4eso/trigonometria/Trigonometria/trigonometria/medidas.htm">«Funciones circulares (trigonométricas): Razones trigonométricas, seno, coseno y tangente. Aplicaciones de medida»</a> <i>www.xtec.cat</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2022-11-11)</small></span></span>.</span> </li> <li id="cite_note-17"><a href="#cite_ref-17">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/pythagoras/PTForReciprocals.shtml">«Pythagorean Theorem for the Reciprocals»</a> <i>www.cut-the-knot.org</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2023-07-13)</small></span></span>.</span> </li> <li id="cite_note-18"><a href="#cite_ref-18">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Gaztelaniaz dago aipu honen iturburua"><b>(Gaztelaniaz)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://www.superprof.es/apuntes/escolar/matematicas/analitica/recta/ecuacion-de-la-recta-que-pasa-por-dos-puntos.html/">«La ecuacion de la recta que pasa por dos puntos | Superprof»</a> <i>Material Didáctico - Superprof</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2022-11-11)</small></span></span>.</span> </li> <li id="cite_note-19"><a href="#cite_ref-19">↑</a> <span class="reference-text"><span class="citation"> </span> <span class="citation"><a rel="nofollow" class="external text" href="https://maitematika.wordpress.com/">«DBHko Matematika»</a> <i>DBHko Matematika</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2022-11-11)</small></span></span>.</span> </li> <li id="cite_note-20"><a href="#cite_ref-20">↑</a> <span class="reference-text"><span class="citation"> </span> <span class="citation"> <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=oYtABjqNpyM"><i>20801 Koordenatu kartesiarrak. </i></a> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2022-11-11)</small></span></span>.</span> </li> <li id="cite_note-21"><a href="#cite_ref-21">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFHoward_Anton;_Chris_Rorres2010">Howard Anton; Chris Rorres.  (2010). <a rel="nofollow" class="external text" href="https://books.google.nl/books?id=1PJ-WHepeBsC&pg=PA336&redir_esc=y#v=onepage&q&f=false"><i>Elementary Linear Algebra: Applications. </i></a> (10. argitaraldia) Wiley, 336 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-470-43205-1" title="Berezi:BookSources/978-0-470-43205-1">978-0-470-43205-1</a>.</span>.</span> </li> <li id="cite_note-22"><a href="#cite_ref-22">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFSaxe2002">Saxe, Karen.  (2002). <a rel="nofollow" class="external text" href="https://books.google.nl/books?id=QALoZC64ea0C&pg=PA7&redir_esc=y#v=onepage&q&f=false"><i>Beginning functional analysis. </i></a> Springer, 7 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/0-387-95224-1" title="Berezi:BookSources/0-387-95224-1">0-387-95224-1</a>.</span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pitagorasen_teorema&veaction=edit&section=24" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pitagorasen_teorema&action=edit&section=24" title="Edit section's source code: Kanpo estekak"><span>aldatu iturburu kodea</span></a><span 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