Losango – Wikipédia, a enciclopédia livre

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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">Início</div> </a> </li> <li id="toc-Etimologia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etimologia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Etimologia</span> </div> </a> <ul id="toc-Etimologia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriedades_geométricas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriedades_geométricas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Propriedades geométricas</span> </div> </a> <button aria-controls="toc-Propriedades_geométricas-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>Alternar a subsecção Propriedades geométricas</span> </button> <ul id="toc-Propriedades_geométricas-sublist" class="vector-toc-list"> <li id="toc-Demonstrações_das_propriedades" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Demonstrações_das_propriedades"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Demonstrações das propriedades</span> </div> </a> <ul id="toc-Demonstrações_das_propriedades-sublist" class="vector-toc-list"> <li id="toc-1°_Propriedade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#1°_Propriedade"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>1° Propriedade</span> </div> </a> <ul id="toc-1°_Propriedade-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2°_Propriedade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2°_Propriedade"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>2° Propriedade</span> </div> </a> <ul id="toc-2°_Propriedade-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3°_Propriedade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#3°_Propriedade"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>3° Propriedade</span> </div> </a> <ul id="toc-3°_Propriedade-sublist" class="vector-toc-list"> <li id="toc-Todo_losango_tem_diagonais_perpendiculares" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Todo_losango_tem_diagonais_perpendiculares"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3.1</span> <span>Todo losango tem diagonais perpendiculares</span> </div> </a> <ul id="toc-Todo_losango_tem_diagonais_perpendiculares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Todo_paralelogramo_que_possui_diagonais_perpendiculares_é_losango" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Todo_paralelogramo_que_possui_diagonais_perpendiculares_é_losango"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3.2</span> <span>Todo paralelogramo que possui diagonais perpendiculares é losango</span> </div> </a> <ul id="toc-Todo_paralelogramo_que_possui_diagonais_perpendiculares_é_losango-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Ângulos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ângulos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Ângulos</span> </div> </a> <ul id="toc-Ângulos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Área" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Área"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Área</span> </div> </a> <ul id="toc-Área-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Incentro" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Incentro"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Incentro</span> </div> </a> <ul id="toc-Incentro-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <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="Alternar o índice" > <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">Alternar o índice</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">Losango</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="Ir para um artigo noutra língua. Disponível em 103 línguas" > <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-103" 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">103 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D9%8A%D9%86_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9)" title="معين (هندسة رياضية) — árabe" lang="ar" hreflang="ar" data-title="معين (هندسة رياضية)" data-language-autonym="العربية" data-language-local-name="árabe" 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%A7%B0%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%BE%E0%A6%9B" title="ৰম্বাছ — assamês" lang="as" hreflang="as" data-title="ৰম্বাছ" data-language-autonym="অসমীয়া" data-language-local-name="assamês" 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/Rombu" title="Rombu — asturiano" lang="ast" hreflang="ast" data-title="Rombu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Romb" title="Romb — azerbaijano" lang="az" hreflang="az" data-title="Romb" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — bashkir" lang="ba" hreflang="ba" data-title="Ромб" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Rombus" title="Rombus — Central Bikol" lang="bcl" hreflang="bcl" data-title="Rombus" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — bielorrusso" lang="be" hreflang="be" data-title="Ромб" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Ромб" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — búlgaro" lang="bg" hreflang="bg" data-title="Ромб" data-language-autonym="Български" data-language-local-name="búlgaro" 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%B0%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%B8" title="রম্বস — bengalês" lang="bn" hreflang="bn" data-title="রম্বস" data-language-autonym="বাংলা" data-language-local-name="bengalês" 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/Lankell" title="Lankell — bretão" lang="br" hreflang="br" data-title="Lankell" data-language-autonym="Brezhoneg" data-language-local-name="bretão" 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/Romb" title="Romb — bósnio" lang="bs" hreflang="bs" data-title="Romb" data-language-autonym="Bosanski" data-language-local-name="bósnio" 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/Rombe" title="Rombe — catalão" lang="ca" hreflang="ca" data-title="Rombe" data-language-autonym="Català" data-language-local-name="catalão" 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/%D9%85%DB%95%D8%B9%DB%8C%D9%86" title="مەعین — curdo central" lang="ckb" hreflang="ckb" data-title="مەعین" data-language-autonym="کوردی" data-language-local-name="curdo central" 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/Rombu" title="Rombu — córsico" lang="co" hreflang="co" data-title="Rombu" data-language-autonym="Corsu" data-language-local-name="córsico" 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/Koso%C4%8Dtverec" title="Kosočtverec — checo" lang="cs" hreflang="cs" data-title="Kosočtverec" data-language-autonym="Čeština" data-language-local-name="checo" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — chuvash" lang="cv" hreflang="cv" data-title="Ромб" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" 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/Rhombws" title="Rhombws — galês" lang="cy" hreflang="cy" data-title="Rhombws" data-language-autonym="Cymraeg" data-language-local-name="galês" 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/Rombe" title="Rombe — dinamarquês" lang="da" hreflang="da" data-title="Rombe" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Raute" title="Raute — alemão" lang="de" hreflang="de" data-title="Raute" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Ruta" title="Ruta — baixo sorábio" lang="dsb" hreflang="dsb" data-title="Ruta" data-language-autonym="Dolnoserbski" data-language-local-name="baixo sorábio" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A1%CF%8C%CE%BC%CE%B2%CE%BF%CF%82" title="Ρόμβος — grego" lang="el" hreflang="el" data-title="Ρόμβος" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Rhombus" title="Rhombus — inglês" lang="en" hreflang="en" data-title="Rhombus" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rombo" title="Rombo — esperanto" lang="eo" hreflang="eo" data-title="Rombo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Rombo" title="Rombo — espanhol" lang="es" hreflang="es" data-title="Rombo" data-language-autonym="Español" data-language-local-name="espanhol" 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/Romb" title="Romb — estónio" lang="et" hreflang="et" data-title="Romb" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erronbo" title="Erronbo — basco" lang="eu" hreflang="eu" data-title="Erronbo" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%84%D9%88%D8%B2%DB%8C" title="لوزی — persa" lang="fa" hreflang="fa" data-title="لوزی" data-language-autonym="فارسی" data-language-local-name="persa" 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/Nelj%C3%A4k%C3%A4s" title="Neljäkäs — finlandês" lang="fi" hreflang="fi" data-title="Neljäkäs" data-language-autonym="Suomi" data-language-local-name="finlandês" 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/Losange" title="Losange — francês" lang="fr" hreflang="fr" data-title="Losange" data-language-autonym="Français" data-language-local-name="francês" 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/R%C3%BCtj_(Geometrii)" title="Rütj (Geometrii) — frísio setentrional" lang="frr" hreflang="frr" data-title="Rütj (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" 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/Rombas" title="Rombas — irlandês" lang="ga" hreflang="ga" data-title="Rombas" data-language-autonym="Gaeilge" data-language-local-name="irlandês" 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/Rombo" title="Rombo — galego" lang="gl" hreflang="gl" data-title="Rombo" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%95%D7%99%D7%9F" title="מעוין — hebraico" lang="he" hreflang="he" data-title="מעוין" data-language-autonym="עברית" data-language-local-name="hebraico" 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%B8%E0%A4%AE%E0%A4%9A%E0%A4%A4%E0%A5%81%E0%A4%B0%E0%A5%8D%E0%A4%AD%E0%A5%81%E0%A4%9C" title="समचतुर्भुज — hindi" lang="hi" hreflang="hi" data-title="समचतुर्भुज" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Romb" title="Romb — croata" lang="hr" hreflang="hr" data-title="Romb" data-language-autonym="Hrvatski" data-language-local-name="croata" 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/K%C3%B3snik" title="Kósnik — alto sorábio" lang="hsb" hreflang="hsb" data-title="Kósnik" data-language-autonym="Hornjoserbsce" data-language-local-name="alto sorábio" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Lozanj" title="Lozanj — haitiano" lang="ht" hreflang="ht" data-title="Lozanj" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rombusz" title="Rombusz — húngaro" lang="hu" hreflang="hu" data-title="Rombusz" data-language-autonym="Magyar" data-language-local-name="húngaro" 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%87%D5%A5%D5%B2%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Շեղանկյուն — arménio" lang="hy" hreflang="hy" data-title="Շեղանկյուն" data-language-autonym="Հայերեն" data-language-local-name="arménio" 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/Rhombo" title="Rhombo — interlíngua" lang="ia" hreflang="ia" data-title="Rhombo" data-language-autonym="Interlingua" data-language-local-name="interlíngua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Belah_ketupat" title="Belah ketupat — indonésio" lang="id" hreflang="id" data-title="Belah ketupat" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/T%C3%ADgull" title="Tígull — islandês" lang="is" hreflang="is" data-title="Tígull" data-language-autonym="Íslenska" data-language-local-name="islandês" 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/Rombo_(geometria)" title="Rombo (geometria) — italiano" lang="it" hreflang="it" data-title="Rombo (geometria)" data-language-autonym="Italiano" data-language-local-name="italiano" 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/%E8%8F%B1%E5%BD%A2" title="菱形 — japonês" lang="ja" hreflang="ja" data-title="菱形" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Sigaran_kupat" title="Sigaran kupat — javanês" lang="jv" hreflang="jv" data-title="Sigaran kupat" data-language-autonym="Jawa" data-language-local-name="javanês" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%9D%E1%83%9B%E1%83%91%E1%83%98" title="რომბი — georgiano" lang="ka" hreflang="ka" data-title="რომბი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — cazaque" lang="kk" hreflang="kk" data-title="Ромб" data-language-autonym="Қазақша" data-language-local-name="cazaque" 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%85%E1%9E%8F%E1%9E%BB%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%9F%E1%9F%92%E1%9E%98%E1%9E%BE" title="ចតុកោណស្មើ — khmer" lang="km" hreflang="km" data-title="ចតុកោណស្មើ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%88%EB%A6%84%EB%AA%A8" title="마름모 — coreano" lang="ko" hreflang="ko" data-title="마름모" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — quirguiz" lang="ky" hreflang="ky" data-title="Ромб" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" 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/Rhombus_(geometria)" title="Rhombus (geometria) — latim" lang="la" hreflang="la" data-title="Rhombus (geometria)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Roet" title="Roet — limburguês" lang="li" hreflang="li" data-title="Roet" data-language-autonym="Limburgs" data-language-local-name="limburguês" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/R%C3%B3mbo" title="Rómbo — lombardo" lang="lmo" hreflang="lmo" data-title="Rómbo" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rombas" title="Rombas — lituano" lang="lt" hreflang="lt" data-title="Rombas" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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/Rombs" title="Rombs — letão" lang="lv" hreflang="lv" data-title="Rombs" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1%D1%81%D1%8C" title="Ромбсь — mocsa" lang="mdf" hreflang="mdf" data-title="Ромбсь" data-language-autonym="Мокшень" data-language-local-name="mocsa" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Miraefadafy" title="Miraefadafy — malgaxe" lang="mg" hreflang="mg" data-title="Miraefadafy" data-language-autonym="Malagasy" data-language-local-name="malgaxe" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — Eastern Mari" lang="mhr" hreflang="mhr" data-title="Ромб" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — macedónio" lang="mk" hreflang="mk" data-title="Ромб" data-language-autonym="Македонски" data-language-local-name="macedónio" 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%B8%E0%B4%AE%E0%B4%AD%E0%B5%81%E0%B4%9C%E0%B4%B8%E0%B4%BE%E0%B4%AE%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B4%82" title="സമഭുജസാമാന്തരികം — malaiala" lang="ml" hreflang="ml" data-title="സമഭുജസാമാന്തരികം" data-language-autonym="മലയാളം" data-language-local-name="malaiala" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — mongol" lang="mn" hreflang="mn" data-title="Ромб" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Rombus" title="Rombus — malaio" lang="ms" hreflang="ms" data-title="Rombus" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Rombo" title="Rombo — mirandês" lang="mwl" hreflang="mwl" data-title="Rombo" data-language-autonym="Mirandés" data-language-local-name="mirandês" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Ruut" title="Ruut — baixo-alemão" lang="nds" hreflang="nds" data-title="Ruut" data-language-autonym="Plattdüütsch" data-language-local-name="baixo-alemão" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ruit_(meetkunde)" title="Ruit (meetkunde) — neerlandês" lang="nl" hreflang="nl" data-title="Ruit (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" 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/Rombe" title="Rombe — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Rombe" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rombe" title="Rombe — norueguês bokmål" lang="nb" hreflang="nb" data-title="Rombe" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" 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/Lausange" title="Lausange — occitano" lang="oc" hreflang="oc" data-title="Lausange" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AE%E0%A8%9A%E0%A9%81%E0%A8%AC%E0%A8%BE%E0%A8%B9%E0%A9%80%E0%A8%86" title="ਸਮਚੁਬਾਹੀਆ — panjabi" lang="pa" hreflang="pa" data-title="ਸਮਚੁਬਾਹੀਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" 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/Romb" title="Romb — polaco" lang="pl" hreflang="pl" data-title="Romb" data-language-autonym="Polski" data-language-local-name="polaco" 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/Romb" title="Romb — Piedmontese" lang="pms" hreflang="pms" data-title="Romb" 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%B9%DB%8C%D9%86" title="معین — Western Punjabi" lang="pnb" hreflang="pnb" data-title="معین" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Puytu" title="Puytu — quíchua" lang="qu" hreflang="qu" data-title="Puytu" data-language-autonym="Runa Simi" data-language-local-name="quíchua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Romb" title="Romb — romeno" lang="ro" hreflang="ro" data-title="Romb" data-language-autonym="Română" data-language-local-name="romeno" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — russo" lang="ru" hreflang="ru" data-title="Ромб" data-language-autonym="Русский" data-language-local-name="russo" 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%A0%D0%BE%D0%BC%D0%B1" title="Ромб — sakha" lang="sah" hreflang="sah" data-title="Ромб" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Rummu" title="Rummu — siciliano" lang="scn" hreflang="scn" data-title="Rummu" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Romb" title="Romb — servo-croata" lang="sh" hreflang="sh" data-title="Romb" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Rhombus" title="Rhombus — Simple English" lang="en-simple" hreflang="en-simple" data-title="Rhombus" 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/Koso%C5%A1tvorec" title="Kosoštvorec — eslovaco" lang="sk" hreflang="sk" data-title="Kosoštvorec" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Romb" title="Romb — esloveno" lang="sl" hreflang="sl" data-title="Romb" data-language-autonym="Slovenščina" data-language-local-name="esloveno" 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/Gonyoina_tsazamativi" title="Gonyoina tsazamativi — shona" lang="sn" hreflang="sn" data-title="Gonyoina tsazamativi" data-language-autonym="ChiShona" data-language-local-name="shona" 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/Qardhaas" title="Qardhaas — somali" lang="so" hreflang="so" data-title="Qardhaas" data-language-autonym="Soomaaliga" data-language-local-name="somali" 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/Rombi" title="Rombi — albanês" lang="sq" hreflang="sq" data-title="Rombi" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — sérvio" lang="sr" hreflang="sr" data-title="Ромб" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Beulahan_kupat" title="Beulahan kupat — sundanês" lang="su" hreflang="su" data-title="Beulahan kupat" data-language-autonym="Sunda" data-language-local-name="sundanês" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Romb" title="Romb — sueco" lang="sv" hreflang="sv" data-title="Romb" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%9A%E0%AE%A4%E0%AF%81%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="சாய்சதுரம் — tâmil" lang="ta" hreflang="ta" data-title="சாய்சதுரம்" data-language-autonym="தமிழ்" data-language-local-name="tâmil" 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%B0%E0%B0%BE%E0%B0%82%E0%B0%AC%E0%B0%B8%E0%B1%8D" title="రాంబస్ — telugu" lang="te" hreflang="te" data-title="రాంబస్" data-language-autonym="తెలుగు" data-language-local-name="telugu" 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%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%82%E0%B8%99%E0%B8%A1%E0%B9%80%E0%B8%9B%E0%B8%B5%E0%B8%A2%E0%B8%81%E0%B8%9B%E0%B8%B9%E0%B8%99" title="รูปสี่เหลี่ยมขนมเปียกปูน — tailandês" lang="th" hreflang="th" data-title="รูปสี่เหลี่ยมขนมเปียกปูน" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Rombus" title="Rombus — tagalo" lang="tl" hreflang="tl" data-title="Rombus" data-language-autonym="Tagalog" data-language-local-name="tagalo" 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/E%C5%9Fkenar_d%C3%B6rtgen" title="Eşkenar dörtgen — turco" lang="tr" hreflang="tr" data-title="Eşkenar dörtgen" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BC%D0%B1" title="Ромб — ucraniano" lang="uk" hreflang="uk" data-title="Ромб" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Romb" title="Romb — usbeque" lang="uz" hreflang="uz" data-title="Romb" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_thoi" title="Hình thoi — vietnamita" lang="vi" hreflang="vi" data-title="Hình thoi" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Rute" title="Rute — West Flemish" lang="vls" hreflang="vls" data-title="Rute" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Lozindje" title="Lozindje — valão" lang="wa" hreflang="wa" data-title="Lozindje" data-language-autonym="Walon" data-language-local-name="valão" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Rhombus" title="Rhombus — waray" lang="war" hreflang="war" data-title="Rhombus" data-language-autonym="Winaray" data-language-local-name="waray" 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/%E8%8F%B1%E5%BD%A2" title="菱形 — wu" lang="wuu" hreflang="wuu" data-title="菱形" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A0%E1%83%9D%E1%83%9B%E1%83%91%E1%83%98" title="რომბი — Mingrelian" lang="xmf" hreflang="xmf" data-title="რომბი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%8F%B1%E5%BD%A2" title="菱形 — chinês" lang="zh" hreflang="zh" data-title="菱形" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%8F%B1%E5%BD%A2" title="菱形 — cantonês" lang="yue" hreflang="yue" data-title="菱形" data-language-autonym="粵語" data-language-local-name="cantonês" 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/Q41159#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <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/Losango" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:Losango" rel="discussion" title="Discussão sobre o conteúdo da página [t]" accesskey="t"><span>Discussão</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Mudar a variante da língua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">português</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Losango"><span>Ler</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Losango&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Losango&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Losango&action=history" title="Edições anteriores desta página. [h]" accesskey="h"><span>Ver histórico</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramentas de página"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramentas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramentas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mais opções" > <div class="vector-menu-heading"> Operações </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Losango"><span>Ler</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Losango&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Losango&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Losango&action=history"><span>Ver histórico</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Geral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_afluentes/Losango" title="Lista de todas as páginas que contêm hiperligações para esta [j]" accesskey="j"><span>Páginas afluentes</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Altera%C3%A7%C3%B5es_relacionadas/Losango" rel="nofollow" title="Mudanças recentes nas páginas para as quais esta contém hiperligações [k]" accesskey="k"><span>Alterações relacionadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pt.wikipedia.org/wiki/Wikipedia:Carregar_ficheiro" title="Carregar ficheiros [u]" accesskey="u"><span>Carregar ficheiro</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Losango&oldid=69675727" title="Hiperligação permanente para esta revisão desta página"><span>Hiperligação permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Losango&action=info" title="Mais informações sobre esta página"><span>Informações da página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Losango&id=69675727&wpFormIdentifier=titleform" title="Informação sobre como citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FLosango"><span>Obter URL encurtado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FLosango"><span>Descarregar código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Livro&bookcmd=book_creator&referer=Losango"><span>Criar um livro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Losango&action=show-download-screen"><span>Descarregar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Losango&printable=yes" title="Versão para impressão desta página [p]" accesskey="p"><span>Versão para impressão</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Noutros projetos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Rhombi" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q41159" title="Hiperligação para o elemento do repositório de dados [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramentas de página"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspeto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspeto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Origem: Wikipédia, a enciclopédia livre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pt" dir="ltr"><div class="hatnote"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/20px-Disambig_grey.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/30px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/40px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> <b>Nota:</b> Para a empresa de soluções financeiras, veja <a href="/wiki/Losango_(empresa)" title="Losango (empresa)">Losango (empresa)</a>.</div> <div class="hatnote"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/20px-Disambig_grey.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/30px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/40px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> <b>Nota:</b> Para carga heráldica, veja <a href="/wiki/Losango_(her%C3%A1ldica)" class="mw-redirect" title="Losango (heráldica)">Losango (heráldica)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Rhombus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombus.png/220px-Rhombus.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/ba/Rhombus.png 1.5x" data-file-width="284" data-file-height="189" /></a><figcaption>Um losango é uma figura formada por quatro lados de igual <a href="/wiki/Comprimento" title="Comprimento">comprimento</a>.</figcaption></figure> <p><b>Losango</b> (◊) é um <a href="/wiki/Quadril%C3%A1tero" title="Quadrilátero">quadrilátero</a> equilátero, ou seja, é um <a href="/wiki/Pol%C3%ADgono" title="Polígono">polígono</a> formado por quatro lados de igual <a href="/wiki/Comprimento" title="Comprimento">comprimento</a>. Um losango é também um <a href="/wiki/Paralelogramo" title="Paralelogramo">paralelogramo</a>. Alguns autores exigem ainda que nenhum dos ângulos do quadrilátero seja reto para que ele seja considerado um losango.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> </p><p>Todo losango é um <a href="/wiki/Paralelogramo" title="Paralelogramo">paralelogramo</a>, e um losango com ângulos retos é um <a href="/wiki/Quadrado" title="Quadrado">quadrado</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> </p><p>Uma superfície cujos limites são um losango, ou semelhantes a um losango, designa-se por <i>superfície rômbica</i>. </p><p>Em <a href="/wiki/Engenharia" title="Engenharia">Engenharia</a> e em <a href="/wiki/F%C3%ADsica" title="Física">Física</a>, a designação "rombo" é mais comum. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etimologia">Etimologia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=1" title="Editar secção: Etimologia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=1" title="Editar código-fonte da secção: Etimologia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O losango também é chamado de <i>rombo</i> e vem do grego "<span lang="el" class="politonico">ῥόμβος</span>" (rhombos), ou seja, algo que gira que deriva do verbo "ρέμβω" (rhembō) que significa <i>voltas e voltas</i>. Já a palavra <i>losango</i> vem do latim arcaico <i>lausa</i>, que designa uma <i>pedra achatada</i> e também derivou na palavra <i>lousa</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Propriedades_geométricas"><span id="Propriedades_geom.C3.A9tricas"></span>Propriedades geométricas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=2" title="Editar secção: Propriedades geométricas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=2" title="Editar código-fonte da secção: Propriedades geométricas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A seguir temos algumas propriedades dos losangos e em seguida suas respectivas demonstrações. </p> <ol><li>Ângulos opostos têm medidas iguais.</li> <li>As suas diagonais são bissetrizes.</li> <li>As suas diagonais são retas perpendiculares.</li> <li>Todo losango tem um círculo inscrito.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Demonstrações_das_propriedades"><span id="Demonstra.C3.A7.C3.B5es_das_propriedades"></span>Demonstrações das propriedades</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=3" title="Editar secção: Demonstrações das propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=3" title="Editar código-fonte da secção: Demonstrações das propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="1°_Propriedade"><span id="1.C2.B0_Propriedade"></span>1° Propriedade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=4" title="Editar secção: 1° Propriedade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=4" title="Editar código-fonte da secção: 1° Propriedade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Essa propriedade é intuitiva e ela parte do fato de que um losango é um caso especial de um paralelogramo. Como todo paralelogramo possui ângulos opostos congruentes, então todo losango também possui ângulos opostos congruentes. </p> <div class="mw-heading mw-heading4"><h4 id="2°_Propriedade"><span id="2.C2.B0_Propriedade"></span>2° Propriedade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=5" title="Editar secção: 2° Propriedade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=5" title="Editar código-fonte da secção: 2° Propriedade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Losango_e_bissetriz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Losango_e_bissetriz.svg/437px-Losango_e_bissetriz.svg.png" decoding="async" width="437" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Losango_e_bissetriz.svg/656px-Losango_e_bissetriz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Losango_e_bissetriz.svg/874px-Losango_e_bissetriz.svg.png 2x" data-file-width="538" data-file-height="360" /></a><figcaption></figcaption></figure> <p>Para demonstrar essa propriedade vamos, inicialmente, escrevê-la em <a href="/wiki/Nota%C3%A7%C3%A3o_matem%C3%A1tica" title="Notação matemática">notação matemática</a>. </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 ABCD\quad {\text{é um losango de diagonais }}\quad {\overline {AC}}\quad {\text{e}}\quad {\overline {BD}}\qquad \Longrightarrow \qquad {\begin{cases}{\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}\\{\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é um losango de diagonais </mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é bissetriz de</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é bissetriz de</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD\quad {\text{é um losango de diagonais }}\quad {\overline {AC}}\quad {\text{e}}\quad {\overline {BD}}\qquad \Longrightarrow \qquad {\begin{cases}{\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}\\{\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902c22de33daafad0164c9ab072f269f7fa29c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:99.867ex; height:7.843ex;" alt="{\displaystyle ABCD\quad {\text{é um losango de diagonais }}\quad {\overline {AC}}\quad {\text{e}}\quad {\overline {BD}}\qquad \Longrightarrow \qquad {\begin{cases}{\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}\\{\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}\end{cases}}}" /></span> </p><p>Por definição temos que todos os lados são congruentes. Assim cada uma das diagonais divide o losango em dois triângulos isósceles. </p><p>Visto que todo losango é um paralelogramo, temos também que suas diagonais se interceptam em seus pontos médios. Assim, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> o ponto de intersecção das diagonais, temos: </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 {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdca62b04446808e2c738932e9e4a32d1de9fc96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.614ex; height:3.009ex;" alt="{\displaystyle {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}" /></span> </p><p>Unindo essas duas implicações temos: </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 {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}},\quad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}\qquad (LLL)\qquad \Longrightarrow \qquad \triangle {AMB}\equiv \triangle {CMB}\equiv \triangle {CMD}\equiv \triangle {AMD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>A</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">(</mo> <mi>L</mi> <mi>L</mi> <mi>L</mi> <mo stretchy="false">)</mo> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>D</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}},\quad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}\qquad (LLL)\qquad \Longrightarrow \qquad \triangle {AMB}\equiv \triangle {CMB}\equiv \triangle {CMD}\equiv \triangle {AMD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee01b5ccdad674e3edb16eb335feb37edff66703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:124.501ex; height:3.509ex;" alt="{\displaystyle {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}},\quad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}\qquad (LLL)\qquad \Longrightarrow \qquad \triangle {AMB}\equiv \triangle {CMB}\equiv \triangle {CMD}\equiv \triangle {AMD}}" /></span> </p><p>Visto que todos esses triângulos são congruentes, temos: </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 A{\hat {B}}M\equiv {M{\hat {B}}C}\equiv {C{\hat {D}}M}\equiv {M{\hat {D}}A}\qquad \Longrightarrow \qquad {\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>M</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>C</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>M</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>A</mi> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é bissetriz de</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\hat {B}}M\equiv {M{\hat {B}}C}\equiv {C{\hat {D}}M}\equiv {M{\hat {D}}A}\qquad \Longrightarrow \qquad {\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8769365adffb0be26a23bca139f4845bf8dbd28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:78.863ex; height:3.843ex;" alt="{\displaystyle A{\hat {B}}M\equiv {M{\hat {B}}C}\equiv {C{\hat {D}}M}\equiv {M{\hat {D}}A}\qquad \Longrightarrow \qquad {\overline {BD}}\quad {\text{é bissetriz de}}\quad {\hat {B}}\quad {\text{e}}\quad {\hat {D}}}" /></span> </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 M{\hat {A}}B\equiv {B{\hat {C}}M}\equiv {M{\hat {C}}D}\equiv {D{\hat {A}}M}\qquad \Longrightarrow \qquad {\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>B</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>M</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>D</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>M</mi> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é bissetriz de</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M{\hat {A}}B\equiv {B{\hat {C}}M}\equiv {M{\hat {C}}D}\equiv {D{\hat {A}}M}\qquad \Longrightarrow \qquad {\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b139c0f120a434c13eaf50253acaf62568f7251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:78.671ex; height:3.843ex;" alt="{\displaystyle M{\hat {A}}B\equiv {B{\hat {C}}M}\equiv {M{\hat {C}}D}\equiv {D{\hat {A}}M}\qquad \Longrightarrow \qquad {\overline {AC}}\quad {\text{é bissetriz de}}\quad {\hat {A}}\quad {\text{e}}\quad {\hat {C}}}" /></span> </p><p>Logo, as diagonais de um losango são bissetrizes de seus respectivos ângulos internos. </p> <div class="mw-heading mw-heading4"><h4 id="3°_Propriedade"><span id="3.C2.B0_Propriedade"></span>3° Propriedade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=6" title="Editar secção: 3° Propriedade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=6" title="Editar código-fonte da secção: 3° Propriedade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Essa propriedade será demonstrada em duas partes (a 'ida' e a 'volta' do teorema). </p> <div class="mw-heading mw-heading5"><h5 id="Todo_losango_tem_diagonais_perpendiculares">Todo losango tem diagonais perpendiculares</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=7" title="Editar secção: Todo losango tem diagonais perpendiculares" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=7" title="Editar código-fonte da secção: Todo losango tem diagonais perpendiculares"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Losangodem1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Losangodem1.png/437px-Losangodem1.png" decoding="async" width="437" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Losangodem1.png/656px-Losangodem1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Losangodem1.png/874px-Losangodem1.png 2x" data-file-width="1553" data-file-height="865" /></a><figcaption></figcaption></figure> <p>Vamos demonstrar que se um quadrilátero é um losango, então ele é tem as diagonais perpendiculares. </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 ABCD\quad {\text{é losango}}\qquad \Longrightarrow \qquad {\overline {AC}}\perp {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é losango</mtext> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊥</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD\quad {\text{é losango}}\qquad \Longrightarrow \qquad {\overline {AC}}\perp {\overline {BD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122469bb87087d930bea0769a8bb98e3e636f1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.003ex; height:3.843ex;" alt="{\displaystyle ABCD\quad {\text{é losango}}\qquad \Longrightarrow \qquad {\overline {AC}}\perp {\overline {BD}}}" /></span> </p><p>Como todo losango é um paralelogramo (por possuir lados opostos congruentes), temos que suas diagonais se interceptam em seus respectivos pontos médios. Ou seja, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69bf8ccc972402f794c261e00b222f396bcce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.692ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}}" /></span> 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 {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71fd62ec7ad127356017d4a62c77e118fd00d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.803ex; height:3.009ex;" alt="{\displaystyle {\overline {BD}}}" /></span> as diagonais temos: </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 M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7226a1b249d2365791a9f714a3a513ad1ccc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:59.619ex; height:3.009ex;" alt="{\displaystyle M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}" /></span> </p><p>Agora tomamos os seguintes triângulos: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle {AMB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {AMB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e793c7a8ec55e4897f3cd42922f9bb11884c2ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.015ex; height:2.176ex;" alt="{\displaystyle \triangle {AMB}}" /></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 \triangle {AMD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {AMD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7cc64ffabe899b3b4e67391d92897cb5e7d2ecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.176ex; height:2.176ex;" alt="{\displaystyle \triangle {AMD}}" /></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 \triangle {CMB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {CMB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4b68379a8874c07055ad70bbfe5ad2b100f8e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.038ex; height:2.176ex;" alt="{\displaystyle \triangle {CMB}}" /></span> 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 \triangle {CMD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {CMD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef9be5cd1e2e98c8b7fc542060c9fb13dc84bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.199ex; height:2.176ex;" alt="{\displaystyle \triangle {CMD}}" /></span>. </p><p>Pelo caso de congruência <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LLL}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>L</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle LLL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b46ddd153614ec9bc44a04c2d56762845893300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.749ex; height:2.176ex;" alt="{\displaystyle LLL}" /></span>, temos as congruências: </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 \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>D</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db979b9cf3fd694d329e64f6d4a6f326bc59914b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:41.723ex; height:2.176ex;" alt="{\displaystyle \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}}" /></span>. </p><p>Assim verificamos que os ângulos de vértices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> são congruentes e suplementares, ou seja: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}\perp {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊥</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}\perp {\overline {BD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8854677eddfc61e6d2dd3476adbce8f085c89693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.594ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}\perp {\overline {BD}}}" /></span> </p> <div class="mw-heading mw-heading5"><h5 id="Todo_paralelogramo_que_possui_diagonais_perpendiculares_é_losango"><span id="Todo_paralelogramo_que_possui_diagonais_perpendiculares_.C3.A9_losango"></span>Todo paralelogramo que possui diagonais perpendiculares é losango</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=8" title="Editar secção: Todo paralelogramo que possui diagonais perpendiculares é losango" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=8" title="Editar código-fonte da secção: Todo paralelogramo que possui diagonais perpendiculares é losango"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Losangodem2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Losangodem2.png/366px-Losangodem2.png" decoding="async" width="366" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Losangodem2.png/549px-Losangodem2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Losangodem2.png/732px-Losangodem2.png 2x" data-file-width="1227" data-file-height="578" /></a><figcaption></figcaption></figure> <p>Vamos demonstrar se um paralelogramo possui diagonais perpendiculares, então ele é um losango. </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 ABCD\quad {\text{é um paralelogramo tal que}}\quad {\overline {AC}}\perp {\overline {BD}}\qquad \Longrightarrow \qquad {ABCD}\quad {\text{é losango}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é um paralelogramo tal que</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊥</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>é losango</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD\quad {\text{é um paralelogramo tal que}}\quad {\overline {AC}}\perp {\overline {BD}}\qquad \Longrightarrow \qquad {ABCD}\quad {\text{é losango}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1a913e6bc516dffa4677b8d0dece0f2fa81599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:83.433ex; height:3.843ex;" alt="{\displaystyle ABCD\quad {\text{é um paralelogramo tal que}}\quad {\overline {AC}}\perp {\overline {BD}}\qquad \Longrightarrow \qquad {ABCD}\quad {\text{é losango}}}" /></span> </p><p>Como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}" /></span> é paralelogramo, temos que seus lados opostos são congruentes, ou seja: </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 {\overline {AB}}\equiv {\overline {CD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}\equiv {\overline {CD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf895353b2d8d8cc2fc625109e5c03619008bf2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.526ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}\equiv {\overline {CD}}}" /></span>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 {\overline {BC}}\equiv {\overline {AD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}\equiv {\overline {AD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29aadea896960d375d4957c4be402e0e8be2de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.594ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}\equiv {\overline {AD}}}" /></span>. </p><p>Também, por ser paralelogramo temos que as diagonais interceptam-se em seus respectivos pontos médios, ou seja: </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 M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7226a1b249d2365791a9f714a3a513ad1ccc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:59.619ex; height:3.009ex;" alt="{\displaystyle M={\overline {AC}}\cap {\overline {BD}}\qquad \Longrightarrow \qquad {\overline {AM}}\equiv {\overline {MC}}\quad {\text{e}}\quad {\overline {BM}}\equiv {\overline {MD}}}" /></span> </p><p>Como diagonais são perpendiculares, temo que : </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Losangodem3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Losangodem3.png/376px-Losangodem3.png" decoding="async" width="376" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Losangodem3.png/564px-Losangodem3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Losangodem3.png/752px-Losangodem3.png 2x" data-file-width="1255" data-file-height="520" /></a><figcaption></figcaption></figure> <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{\hat {M}}B\equiv {B{\hat {M}}C}\equiv {C{\hat {M}}D}\equiv {D{\hat {M}}A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>B</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>C</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>D</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\hat {M}}B\equiv {B{\hat {M}}C}\equiv {C{\hat {M}}D}\equiv {D{\hat {M}}A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1a45c1be8c92dc72fc9698da309111551e04cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:33.46ex; height:2.843ex;" alt="{\displaystyle A{\hat {M}}B\equiv {B{\hat {M}}C}\equiv {C{\hat {M}}D}\equiv {D{\hat {M}}A}}" /></span>. </p><p>Dessas relações, através do caso de congruência <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LAL}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>A</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle LAL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84eb17e46807b5e0dec4d9fa6320f9f49db6adbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.909ex; height:2.176ex;" alt="{\displaystyle LAL}" /></span>, temos: </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 \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}\qquad \Longrightarrow \qquad {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>M</mi> <mi>D</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>B</mi> </mrow> <mo>≡</mo> <mi mathvariant="normal">△</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>M</mi> <mi>D</mi> </mrow> <mspace width="2em"></mspace> <mo stretchy="false">⟹</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>A</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}\qquad \Longrightarrow \qquad {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1474bbcc001ad0891a2e8afcdde5c9fba567863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:80.328ex; height:3.009ex;" alt="{\displaystyle \triangle {AMB}\equiv \triangle {AMD}\equiv \triangle {CMB}\equiv \triangle {CMD}\qquad \Longrightarrow \qquad {\overline {AB}}\equiv {\overline {BC}}\equiv {\overline {CD}}\equiv {\overline {DA}}}" /></span>. </p><p>Logo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}" /></span> é um losango.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>4<span>]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ângulos"><span id=".C3.82ngulos"></span>Ângulos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=9" title="Editar secção: Ângulos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=9" title="Editar código-fonte da secção: Ângulos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O único losango que não possui dois ângulos agudos (menores que 90°) e dois ângulos obtusos (maiores que 90°) é o <a href="/wiki/Quadrado" title="Quadrado">quadrado</a>. O quadrado possui quatro ângulos iguais a 90°. </p> <div class="mw-heading mw-heading3"><h3 id="Área"><span id=".C3.81rea"></span>Área</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=10" title="Editar secção: Área" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=10" title="Editar código-fonte da secção: Área"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Rhombus1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Rhombus1.svg/280px-Rhombus1.svg.png" decoding="async" width="280" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Rhombus1.svg/420px-Rhombus1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Rhombus1.svg/560px-Rhombus1.svg.png 2x" data-file-width="539" data-file-height="332" /></a><figcaption>A altura <i>h</i> é a distância perpendicular entre dois lados quaisquer não-adjacentes, ou o diâmetro do <a href="/w/index.php?title=Circulo_inscrito&action=edit&redlink=1" class="new" title="Circulo inscrito (página não existe)">circulo inscrito</a> (<i>h</i>=2<i>r</i>).</figcaption></figure> <p>Existem varias formas de se visualizar a fórmula da <a href="/wiki/%C3%81rea" title="Área">área</a>. A mais usual é partindo do que se conhece sobre a fórmula da área do <a href="/wiki/Paralelogramo" title="Paralelogramo">paralelogramo</a>. A área <i>A</i> de um paralelogramo é o produto da sua base pela sua altura (<i>h</i> na ilustração). No losango qualquer lado, todos de comprimento <i>a</i>, se presta a fazer o papel de base: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=a\cdot h=a\cdot 2r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>h</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mn>2</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=a\cdot h=a\cdot 2r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757a4d055b246aed64b241602eb10eb7703fc87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.308ex; height:2.176ex;" alt="{\displaystyle A=a\cdot h=a\cdot 2r}" /></span></dd></dl> <p>Outra forma usual de visualizar a área do losango, é percebendo que o traçado de suas diagonais permite dividi-lo em quatro <a href="/wiki/Tri%C3%A2ngulo_ret%C3%A2ngulo" title="Triângulo retângulo">triângulos retângulos</a> simétricos e de mesma área: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\cdot K={\frac {p\cdot q}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\cdot K={\frac {p\cdot q}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5925262900ab79ebe3e0970132d47ef175faea0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.602ex; height:4.843ex;" alt="{\displaystyle A=4\cdot K={\frac {p\cdot q}{2}}}" /></span></dd></dl> <p>onde <i>K</i> é a área de um dos triângulos, <i>p</i> e <i>q</i> as diagonais do losango.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>5<span>]</span></a></sup> A correspondência com a área da primeira abordagem é demonstrada na <a href="#Incentro">análise do incentro</a>. </p><p>Uma terceira forma, se baseia na primeira, mas expressando <i>h</i> como projeção de um lado <i>a</i>, ou seja, como cateto oposto do ângulo <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>, portando expressando através do seno, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=a^{2}\cdot \sin \alpha =a^{2}\cdot \sin \beta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=a^{2}\cdot \sin \alpha =a^{2}\cdot \sin \beta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84222f76717cebf6c0bf7e8d476b61408f9dde95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.818ex; height:3.009ex;" alt="{\displaystyle A=a^{2}\cdot \sin \alpha =a^{2}\cdot \sin \beta ,}" /></span></dd></dl> <p>onde é lembrado que o seno de qualquer ângulo do losango é o mesmo (num paralelogramo os ângulos são suplementares entre si). </p> <div class="mw-heading mw-heading3"><h3 id="Incentro">Incentro</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Losango&veaction=edit&section=11" title="Editar secção: Incentro" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Losango&action=edit&section=11" title="Editar código-fonte da secção: Incentro"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Rombo.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/8e/Rombo.png" decoding="async" width="158" height="218" class="mw-file-element" data-file-width="158" data-file-height="218" /></a><figcaption>Visualizando o losango como quadra triângulos-retângulo simétricos (de área <i>K</i>), dados pelas diagonais (<i>p</i> e <i>q</i>).</figcaption></figure> <p>Para calcular o raio do incentro de um losango, basta usar a seguinte fórmula considerando <i>p</i> e <i>q</i> como diagonais dele. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {p\cdot q}{2{\sqrt {p^{2}+q^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {p\cdot q}{2{\sqrt {p^{2}+q^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d48823306e6d3bfb51f1a160319c30430d5f9d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.314ex; height:6.176ex;" alt="{\displaystyle r={\frac {p\cdot q}{2{\sqrt {p^{2}+q^{2}}}}}.}" /></span></dd></dl> <p>observando o triângulo-retângulo de hipotenusa <i>a</i> e catetos <i>p/2</i> e <i>q/2</i>, concluímos que a área <i>K</i> de cada triângulo ilustrado é: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=a\cdot r/2={\frac {p\cdot q}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> </mrow> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=a\cdot r/2={\frac {p\cdot q}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e139283124d6f1abbd5f84a7bb5403c77ab04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.3ex; height:4.843ex;" alt="{\displaystyle K=a\cdot r/2={\frac {p\cdot q}{8}}}" /></span></dd></dl> <p>Também pode ser aplicada a seguinte fórmula para o cálculo da área de um losango: </p><p>Área=D*d/2 </p><p>D representa o comprimento de sua diagonal maior e d representa o comprimento de sua diagonal menor. </p><p>Ou seja, a área de um losango é a metade do produto de suas diagonais. </p> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Campos (1735), <span class="plainlinks"><a rel="nofollow" class="external text" href="http://books.google.com/books?id=Hof-Xq5rm5kC&pg=PA6&dq=%22Rhombo%22">p. 6</a></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Nota: a definição original de <a href="/wiki/Euclides" title="Euclides">Euclides</a> e de alguns dicionários de língua inglesa excluem os quadrados, mas os matemáticos modernos preferem a definição inclusiva.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Square"><cite id="CITEREFWeisstein" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Square.html">«Square»</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (em inglês)</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ALosango&rft.atitle=Square&rft.au=Weisstein%2C+Eric+W.&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSquare.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> uso inclusivo</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book">Dolce, Osvaldo (2013). <i>Fundamentos de Matemática Elementar 9: Geometria plana</i>. São Paulo: Atual</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ALosango&rft.aufirst=Osvaldo&rft.aulast=Dolce&rft.btitle=Fundamentos+de+Matem%C3%A1tica+Elementar+9%3A+Geometria+plana&rft.date=2013&rft.genre=book&rft.place=S%C3%A3o+Paulo&rft.pub=Atual&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite class="citation book">Campos, Manoel de (1735). <a rel="nofollow" class="external text" href="http://books.google.com.br/books?id=Hof-Xq5rm5kC"><i>Elementos de geometria plana e solida segundo a ordem de Euclides</i></a>. [S.l.]: Officina Rita-Cassiana</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ALosango&rft.aufirst=Manoel+de&rft.aulast=Campos&rft.btitle=Elementos+de+geometria+plana+e+solida+segundo+a+ordem+de+Euclides&rft.date=1735&rft.genre=book&rft.pub=Officina+Rita-Cassiana&rft_id=http%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DHof-Xq5rm5kC&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Losango&oldid=69675727">https://pt.wikipedia.org/w/index.php?title=Losango&oldid=69675727</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:Pol%C3%ADgonos" title="Categoria:Polígonos">Polígonos</a></li><li><a href="/wiki/Categoria:Desenho_geom%C3%A9trico" title="Categoria:Desenho geométrico">Desenho geométrico</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:!CS1_ingl%C3%AAs-fontes_em_l%C3%ADngua_(en)" title="Categoria:!CS1 inglês-fontes em língua (en)">!CS1 inglês-fontes em língua (en)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 22h59min de 4 de março de 2025.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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