Quadrilateral - Wikipedia

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</div> </a> <ul id="toc-Convex_quadrilateral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Concave_quadrilaterals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Concave_quadrilaterals"> <div class="vector-toc-text"> 1.2 Concave quadrilaterals </div> </a> <ul id="toc-Concave_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_quadrilaterals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_quadrilaterals"> <div class="vector-toc-text"> 2 Complex quadrilaterals </div> </a> <ul id="toc-Complex_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_line_segments" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Special_line_segments"> <div class="vector-toc-text"> 3 Special line segments </div> </a> <ul id="toc-Special_line_segments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_of_a_convex_quadrilateral" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Area_of_a_convex_quadrilateral"> <div class="vector-toc-text"> 4 Area of a convex quadrilateral </div> </a> <button aria-controls="toc-Area_of_a_convex_quadrilateral-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Area of a convex quadrilateral subsection </button> <ul id="toc-Area_of_a_convex_quadrilateral-sublist" class="vector-toc-list"> <li id="toc-Trigonometric_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trigonometric_formulas"> <div class="vector-toc-text"> 4.1 Trigonometric formulas </div> </a> <ul id="toc-Trigonometric_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-trigonometric_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-trigonometric_formulas"> <div class="vector-toc-text"> 4.2 Non-trigonometric formulas </div> </a> <ul id="toc-Non-trigonometric_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_formulas"> <div class="vector-toc-text"> 4.3 Vector formulas </div> </a> <ul id="toc-Vector_formulas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diagonals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Diagonals"> <div class="vector-toc-text"> 5 Diagonals </div> </a> <button aria-controls="toc-Diagonals-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Diagonals subsection </button> <ul id="toc-Diagonals-sublist" class="vector-toc-list"> <li id="toc-Properties_of_the_diagonals_in_quadrilaterals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_of_the_diagonals_in_quadrilaterals"> <div class="vector-toc-text"> 5.1 Properties of the diagonals in quadrilaterals </div> </a> <ul id="toc-Properties_of_the_diagonals_in_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lengths_of_the_diagonals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lengths_of_the_diagonals"> <div class="vector-toc-text"> 5.2 Lengths of the diagonals </div> </a> <ul id="toc-Lengths_of_the_diagonals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations_of_the_parallelogram_law_and_Ptolemy's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations_of_the_parallelogram_law_and_Ptolemy's_theorem"> <div class="vector-toc-text"> 5.3 Generalizations of the parallelogram law and Ptolemy's theorem </div> </a> <ul id="toc-Generalizations_of_the_parallelogram_law_and_Ptolemy's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_metric_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_metric_relations"> <div class="vector-toc-text"> 5.4 Other metric relations </div> </a> <ul id="toc-Other_metric_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angle_bisectors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angle_bisectors"> <div class="vector-toc-text"> 6 Angle bisectors </div> </a> <ul id="toc-Angle_bisectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bimedians" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bimedians"> <div class="vector-toc-text"> 7 Bimedians </div> </a> <ul id="toc-Bimedians-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trigonometric_identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Trigonometric_identities"> <div class="vector-toc-text"> 8 Trigonometric identities </div> </a> <ul id="toc-Trigonometric_identities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inequalities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inequalities"> <div class="vector-toc-text"> 9 Inequalities </div> </a> <button aria-controls="toc-Inequalities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Inequalities subsection </button> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> <li id="toc-Area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Area"> <div class="vector-toc-text"> 9.1 Area </div> </a> <ul id="toc-Area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonals_and_bimedians" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonals_and_bimedians"> <div class="vector-toc-text"> 9.2 Diagonals and bimedians </div> </a> <ul id="toc-Diagonals_and_bimedians-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sides" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sides"> <div class="vector-toc-text"> 9.3 Sides </div> </a> <ul id="toc-Sides-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Maximum_and_minimum_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Maximum_and_minimum_properties"> <div class="vector-toc-text"> 10 Maximum and minimum properties </div> </a> <ul id="toc-Maximum_and_minimum_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Remarkable_points_and_lines_in_a_convex_quadrilateral" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Remarkable_points_and_lines_in_a_convex_quadrilateral"> <div class="vector-toc-text"> 11 Remarkable points and lines in a convex quadrilateral </div> </a> <ul id="toc-Remarkable_points_and_lines_in_a_convex_quadrilateral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_properties_of_convex_quadrilaterals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_properties_of_convex_quadrilaterals"> <div class="vector-toc-text"> 12 Other properties of convex quadrilaterals </div> </a> <ul id="toc-Other_properties_of_convex_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Taxonomy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Taxonomy"> <div class="vector-toc-text"> 13 Taxonomy </div> </a> <ul id="toc-Taxonomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skew_quadrilaterals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Skew_quadrilaterals"> <div class="vector-toc-text"> 14 Skew quadrilaterals </div> </a> <ul id="toc-Skew_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 15 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 16 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <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" > Toggle the table of contents </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">Quadrilateral</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="Go to an article in another language. Available in 103 languages" > <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" > 103 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%9F%D0%BB%D3%80%D0%B8%D0%BC%D1%8D" title="ПлӀимэ – Kabardian" lang="kbd" hreflang="kbd" data-title="ПлӀимэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target">Адыгэбзэ</a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vierhoek" title="Vierhoek – Afrikaans" lang="af" hreflang="af" data-title="Vierhoek" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%AB%E1%89%B5_%E1%88%9B%E1%8B%95%E1%8B%98%E1%8A%95" title="አራት ማዕዘን – Amharic" lang="am" hreflang="am" data-title="አራት ማዕዘን" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/F%C4%93owerecge" title="Fēowerecge – Old English" lang="ang" hreflang="ang" data-title="Fēowerecge" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%A8%D8%A7%D8%B9%D9%8A_%D8%A3%D8%B6%D9%84%D8%A7%D8%B9" title="رباعي أضلاع – Arabic" lang="ar" hreflang="ar" data-title="رباعي أضلاع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9A%E0%A6%A4%E0%A7%81%E0%A7%B0%E0%A7%8D%E0%A6%AD%E0%A7%81%E0%A6%9C" title="চতুৰ্ভুজ – Assamese" lang="as" hreflang="as" data-title="চতুৰ্ভুজ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cuadril%C3%A1teru" title="Cuadriláteru – Asturian" lang="ast" hreflang="ast" data-title="Cuadriláteru" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/D%C3%B6rdbucaql%C4%B1" title="Dördbucaqlı – Azerbaijani" lang="az" hreflang="az" data-title="Dördbucaqlı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A6%A4%E0%A7%81%E0%A6%B0%E0%A7%8D%E0%A6%AD%E0%A7%81%E0%A6%9C" title="চতুর্ভুজ – Bangla" lang="bn" hreflang="bn" data-title="চতুর্ভুজ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%AC-kak-h%C3%AAng" title="Sì-kak-hêng – Minnan" lang="nan" hreflang="nan" data-title="Sì-kak-hêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%94%D2%AF%D1%80%D1%82%D0%BC%D3%A9%D0%B9%D3%A9%D1%88" title="Дүртмөйөш – Bashkir" lang="ba" hreflang="ba" data-title="Дүртмөйөш" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A7%D0%B0%D1%82%D1%8B%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA" title="Чатырохвугольнік – Belarusian" lang="be" hreflang="be" data-title="Чатырохвугольнік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A7%D0%B0%D1%82%D1%8B%D1%80%D0%BE%D1%85%D0%BA%D1%83%D1%82%D0%BD%D1%96%D0%BA" title="Чатырохкутнік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Чатырохкутнік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Kwadrilatero" title="Kwadrilatero – Central Bikol" lang="bcl" hreflang="bcl" data-title="Kwadrilatero" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%B8%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA" title="Четириъгълник – Bulgarian" lang="bg" hreflang="bg" data-title="Четириъгълник" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/%C4%8Cetverougao" title="Četverougao – Bosnian" lang="bs" hreflang="bs" data-title="Četverougao" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Pevarc%27hostezeg" title="Pevarc'hostezeg – Breton" lang="br" hreflang="br" data-title="Pevarc'hostezeg" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quadril%C3%A0ter" title="Quadrilàter – Catalan" lang="ca" hreflang="ca" data-title="Quadrilàter" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D0%B2%D0%B0%D1%82%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%C4%95%D1%85" title="Тăваткĕтеслĕх – Chuvash" lang="cv" hreflang="cv" data-title="Тăваткĕтеслĕх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8Cty%C5%99%C3%BAheln%C3%ADk" title="Čtyřúhelník – Czech" lang="cs" hreflang="cs" data-title="Čtyřúhelník" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gonyoina" title="Gonyoina – Shona" lang="sn" hreflang="sn" data-title="Gonyoina" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pedrochr" title="Pedrochr – Welsh" lang="cy" hreflang="cy" data-title="Pedrochr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Firkant" title="Firkant – Danish" lang="da" hreflang="da" data-title="Firkant" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Njealje%C4%8Diegat" title="Njealječiegat – Northern Sami" lang="se" hreflang="se" data-title="Njealječiegat" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target">Davvisámegiella</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Viereck" title="Viereck – German" lang="de" hreflang="de" data-title="Viereck" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Nelinurk" title="Nelinurk – Estonian" lang="et" hreflang="et" data-title="Nelinurk" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%AC%CF%80%CE%BB%CE%B5%CF%85%CF%81%CE%BF" title="Τετράπλευρο – Greek" lang="el" hreflang="el" data-title="Τετράπλευρο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuadril%C3%A1tero" title="Cuadrilátero – Spanish" lang="es" hreflang="es" data-title="Cuadrilátero" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvarlatero" title="Kvarlatero – Esperanto" lang="eo" hreflang="eo" data-title="Kvarlatero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lauki" title="Lauki – Basque" lang="eu" hreflang="eu" data-title="Lauki" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%87%D8%A7%D8%B1%D8%B6%D9%84%D8%B9%DB%8C" title="چهارضلعی – Persian" lang="fa" hreflang="fa" data-title="چهارضلعی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Quadrilat%C3%A8re" title="Quadrilatère – French" lang="fr" hreflang="fr" data-title="Quadrilatère" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Ceithir-che%C3%A0rnach" title="Ceithir-cheàrnach – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Ceithir-cheàrnach" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cuadril%C3%A1tero" title="Cuadrilátero – Galician" lang="gl" hreflang="gl" data-title="Cuadrilátero" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EA%B0%81%ED%98%95" title="사각형 – Korean" lang="ko" hreflang="ko" data-title="사각형" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%A1%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Քառանկյուն – Armenian" lang="hy" hreflang="hy" data-title="Քառանկյուն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%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="चतुर्भुज – Hindi" lang="hi" hreflang="hi" data-title="चतुर्भुज" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/%C4%8Cetverokut" title="Četverokut – Croatian" lang="hr" hreflang="hr" data-title="Četverokut" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Quadrilatero" title="Quadrilatero – Ido" lang="io" hreflang="io" data-title="Quadrilatero" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Segi_empat" title="Segi empat – Indonesian" lang="id" hreflang="id" data-title="Segi empat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Quadrilatero" title="Quadrilatero – Interlingua" lang="ia" hreflang="ia" data-title="Quadrilatero" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferhyrningur" title="Ferhyrningur – Icelandic" lang="is" hreflang="is" data-title="Ferhyrningur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quadrilatero" title="Quadrilatero – Italian" lang="it" hreflang="it" data-title="Quadrilatero" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%95%D7%91%D7%A2" title="מרובע – Hebrew" lang="he" hreflang="he" data-title="מרובע" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Quadrilateral" title="Quadrilateral – Javanese" lang="jv" hreflang="jv" data-title="Quadrilateral" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9D%E1%83%97%E1%83%AE%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94%E1%83%93%E1%83%98" title="ოთხკუთხედი – Georgian" lang="ka" hreflang="ka" data-title="ოთხკუთხედი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D1%82%D0%B1%D2%B1%D1%80%D1%8B%D1%88" title="Төртбұрыш – Kazakh" lang="kk" hreflang="kk" data-title="Төртбұрыш" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Pembenne" title="Pembenne – Swahili" lang="sw" hreflang="sw" data-title="Pembenne" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C3%87argo%C5%9Fe" title="Çargoşe – Kurdish" lang="ku" hreflang="ku" data-title="Çargoşe" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Quadrilaterale" title="Quadrilaterale – Latin" lang="la" hreflang="la" data-title="Quadrilaterale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%8Cetrst%C5%ABris" title="Četrstūris – Latvian" lang="lv" hreflang="lv" data-title="Četrstūris" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Keturkampis" title="Keturkampis – Lithuanian" lang="lt" hreflang="lt" data-title="Keturkampis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Veerhook" title="Veerhook – Limburgish" lang="li" hreflang="li" data-title="Veerhook" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Quadril%C3%A0ter_(geometr%C3%ACa)" title="Quadrilàter (geometrìa) – Lombard" lang="lmo" hreflang="lmo" data-title="Quadrilàter (geometrìa)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gysz%C3%B6g" title="Négyszög – Hungarian" lang="hu" hreflang="hu" data-title="Négyszög" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%B8%D1%80%D0%B8%D0%B0%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Четириаголник – Macedonian" lang="mk" hreflang="mk" data-title="Четириаголник" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Efadafy" title="Efadafy – Malagasy" lang="mg" hreflang="mg" data-title="Efadafy" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9A%E0%B4%A4%E0%B5%81%E0%B5%BC%E0%B4%AD%E0%B5%81%E0%B4%9C%E0%B4%82" title="ചതുർഭുജം – Malayalam" lang="ml" hreflang="ml" data-title="ചതുർഭുജം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%9A%E0%A5%8C%E0%A4%95%E0%A5%8B%E0%A4%A8" title="चौकोन – Marathi" lang="mr" hreflang="mr" data-title="चौकोन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Segi_empat" title="Segi empat – Malay" lang="ms" hreflang="ms" data-title="Segi empat" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%90%E1%80%AF%E1%80%82%E1%80%B6" title="စတုဂံ – Burmese" lang="my" hreflang="my" data-title="စတုဂံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vierhoek" title="Vierhoek – Dutch" lang="nl" hreflang="nl" data-title="Vierhoek" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%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="चतुर्भुज – Nepali" lang="ne" hreflang="ne" data-title="चतुर्भुज" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%AF%E0%A4%95%E0%A5%81%E0%A4%82" title="प्यकुं – Newari" lang="new" hreflang="new" data-title="प्यकुं" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%9B%E8%A7%92%E5%BD%A2" title="四角形 – Japanese" lang="ja" hreflang="ja" data-title="四角形" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Firkant" title="Firkant – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Firkant" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Firkant" title="Firkant – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Firkant" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Quadrilat%C3%A8r" title="Quadrilatèr – Occitan" lang="oc" hreflang="oc" data-title="Quadrilatèr" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%9A%E0%AC%A4%E0%AD%81%E0%AC%B0%E0%AD%8D%E0%AC%AD%E0%AD%81%E0%AC%9C" title="ଚତୁର୍ଭୁଜ – Odia" lang="or" hreflang="or" data-title="ଚତୁର୍ଭୁଜ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/To%CA%BBrtburchak" title="Toʻrtburchak – Uzbek" lang="uz" hreflang="uz" data-title="Toʻrtburchak" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%81%E0%A8%AC%E0%A8%BE%E0%A8%B9%E0%A9%80%E0%A8%86" title="ਚੁਬਾਹੀਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਚੁਬਾਹੀਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%86%D9%88%DA%A9%D9%88%D8%B1" title="چوکور – Western Punjabi" lang="pnb" hreflang="pnb" data-title="چوکور" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%85%D9%84%D9%88%D8%B1%DA%85%D9%86%DA%89%DB%8C" title="څلورڅنډی – Pashto" lang="ps" hreflang="ps" data-title="څلورڅنډی" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</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" title="ចតុកោណ – Khmer" lang="km" hreflang="km" data-title="ចតុកោណ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Czworok%C4%85t" title="Czworokąt – Polish" lang="pl" hreflang="pl" data-title="Czworokąt" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quadril%C3%A1tero" title="Quadrilátero – Portuguese" lang="pt" hreflang="pt" data-title="Quadrilátero" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Patrulater" title="Patrulater – Romanian" lang="ro" hreflang="ro" data-title="Patrulater" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Tawak%27uchu" title="Tawak'uchu – Quechua" lang="qu" hreflang="qu" data-title="Tawak'uchu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D1%8B%D1%80%D1%91%D1%85%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Четырёхугольник – Russian" lang="ru" hreflang="ru" data-title="Четырёхугольник" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quadrilateral" title="Quadrilateral – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quadrilateral" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%86%D9%88%DA%AA%D9%86%DA%8A%D9%88" title="چوڪنڊو – Sindhi" lang="sd" hreflang="sd" data-title="چوڪنڊو" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C5%A0tvoruholn%C3%ADk" title="Štvoruholník – Slovak" lang="sk" hreflang="sk" data-title="Štvoruholník" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/%C5%A0tirikotnik" title="Štirikotnik – Slovenian" lang="sl" hreflang="sl" data-title="Štirikotnik" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%86%D9%88%D8%A7%D8%B1%D9%84%D8%A7" title="چوارلا – Central Kurdish" lang="ckb" hreflang="ckb" data-title="چوارلا" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE" title="Четвороугао – Serbian" lang="sr" hreflang="sr" data-title="Четвороугао" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/%C4%8Cetverokut" title="Četverokut – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Četverokut" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Nelikulmio" title="Nelikulmio – Finnish" lang="fi" hreflang="fi" data-title="Nelikulmio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fyrh%C3%B6rning" title="Fyrhörning – Swedish" lang="sv" hreflang="sv" data-title="Fyrhörning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Polygon with four sides and four corners</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about four-sided mathematical shapes. For other uses, see <a href="/wiki/Quadrilateral_(disambiguation)" class="mw-disambig" title="Quadrilateral (disambiguation)">Quadrilateral (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Tetragon" redirects here. For the edible plant, see <a href="/wiki/Tetragonia_tetragonioides" title="Tetragonia tetragonioides">Tetragonia tetragonioides</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Quadrilateral</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Six_Quadrilaterals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Six_Quadrilaterals.svg/220px-Six_Quadrilaterals.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Six_Quadrilaterals.svg/330px-Six_Quadrilaterals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Six_Quadrilaterals.svg/440px-Six_Quadrilaterals.svg.png 2x" data-file-width="300" data-file-height="300" /></a><div class="infobox-caption">Some types of quadrilaterals</div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a> and <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a></th><td class="infobox-data">{4} (for square)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Area" title="Area">Area</a></th><td class="infobox-data">various methods; <a href="#Area_of_a_convex_quadrilateral">see below</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Internal_angle" class="mw-redirect" title="Internal angle">Internal angle</a> (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data">90° (for square and rectangle)</td></tr></tbody></table> In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">geometry</a> a quadrilateral is a four-sided <a href="/wiki/Polygon" title="Polygon">polygon</a>, having four <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> (sides) and four <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">corners</a> (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. <a href="/wiki/Pentagon" title="Pentagon">pentagon</a>). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is sometimes denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square ABCD}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f54153304ec3963272dedf0701611bf6633aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.006ex; height:2.176ex;" alt="{\displaystyle \square ABCD}">.<a href="#cite_note-:0-1">[1]</a> Quadrilaterals are either <a href="/wiki/Simple_polygon" title="Simple polygon">simple</a> (not self-intersecting), or <a href="/wiki/Complex_polygon" title="Complex polygon">complex</a> (self-intersecting, or crossed). Simple quadrilaterals are either <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> or <a href="/wiki/Concave_polygon" title="Concave polygon">concave</a>. The <a href="/wiki/Internal_and_external_angle" class="mw-redirect" title="Internal and external angle">interior angles</a> of a simple (and <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planar</a>) quadrilateral ABCD add up to 360 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, that is<a href="#cite_note-:0-1">[1]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mo>=</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea3c18f6a825eb8a3f7e2ccc2c347a7e6c977c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.718ex; height:2.509ex;" alt="{\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ }.}"></dd></dl> This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180° (here, n=4).<a href="#cite_note-2">[2]</a> All non-self-crossing quadrilaterals <a href="/wiki/Tessellation" title="Tessellation">tile the plane</a>, by repeated rotation around the midpoints of their edges.<a href="#cite_note-3">[3]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Simple_quadrilaterals">Simple quadrilaterals</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=1" title="Edit section: Simple quadrilaterals">edit</a>]</div> Any quadrilateral that is not self-intersecting is a simple quadrilateral. <div class="mw-heading mw-heading3"><h3 id="Convex_quadrilateral">Convex quadrilateral</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=2" title="Edit section: Convex quadrilateral">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Euler_diagram_of_quadrilateral_types.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Euler_diagram_of_quadrilateral_types.svg/300px-Euler_diagram_of_quadrilateral_types.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Euler_diagram_of_quadrilateral_types.svg/450px-Euler_diagram_of_quadrilateral_types.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Euler_diagram_of_quadrilateral_types.svg/600px-Euler_diagram_of_quadrilateral_types.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption><a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetries_of_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Symmetries_of_square.svg/300px-Symmetries_of_square.svg.png" decoding="async" width="300" height="341" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Symmetries_of_square.svg/450px-Symmetries_of_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Symmetries_of_square.svg/600px-Symmetries_of_square.svg.png 2x" data-file-width="606" data-file-height="688" /></a><figcaption>Convex quadrilaterals by symmetry, represented with a <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a>.</figcaption></figure> In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. <ul><li>Irregular quadrilateral (<a href="/wiki/British_English" title="British English">British English</a>) or trapezium (<a href="/wiki/North_American_English" title="North American English">North American English</a>): no sides are parallel. (In British English, this was once called a trapezoid. For more, see <a href="/wiki/Trapezoid#Trapezium_vs_Trapezoid" title="Trapezoid">Trapezoid § Trapezium vs Trapezoid</a>.)</li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezium</a> (UK) or <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> (US): at least one pair of opposite sides are <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a>. Trapezia (UK) and trapezoids (US) include parallelograms.</li> <li><a href="/wiki/Isosceles_trapezium" class="mw-redirect" title="Isosceles trapezium">Isosceles trapezium</a> (UK) or <a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoid</a> (US): one pair of opposite sides are parallel and the base <a href="/wiki/Angle" title="Angle">angles</a> are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.</li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a>: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.</li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a>, rhomb:<a href="#cite_note-:0-1">[1]</a> all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).</li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a>: a parallelogram in which adjacent sides are of unequal lengths, and some angles are <a href="/wiki/Angle#Types_of_angles" title="Angle">oblique</a> (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus.<a href="#cite_note-4">[4]</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a>: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).</li> <li><a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">Square</a> (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).</li> <li><a href="https://en.wiktionary.org/wiki/oblong" class="extiw" title="wikt:oblong">Oblong</a>: longer than wide, or wider than long (i.e., a rectangle that is not a square).<a href="#cite_note-5">[5]</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a>: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into <a href="/wiki/Congruent_triangles" class="mw-redirect" title="Congruent triangles">congruent triangles</a>, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.</li></ul> <a href="/wiki/File:Quadrilaterals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Quadrilaterals.svg/661px-Quadrilaterals.svg.png" decoding="async" width="661" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Quadrilaterals.svg/992px-Quadrilaterals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Quadrilaterals.svg/1322px-Quadrilaterals.svg.png 2x" data-file-width="661" data-file-height="320" /></a> <ul><li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential quadrilateral</a>: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.</li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a>: a trapezoid where the four sides are <a href="/wiki/Tangent" title="Tangent">tangents</a> to an <a href="/wiki/Inscribed_circle" class="mw-redirect" title="Inscribed circle">inscribed circle</a>.</li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic quadrilateral</a>: the four vertices lie on a <a href="/wiki/Circumscribed_circle" title="Circumscribed circle">circumscribed circle</a>. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.</li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a>: a kite with two opposite right angles. It is a type of cyclic quadrilateral.</li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic quadrilateral</a>: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal.</li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric quadrilateral</a>: it is both tangential and cyclic.</li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal quadrilateral</a>: the diagonals cross at <a href="/wiki/Right_angle" title="Right angle">right angles</a>.</li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal quadrilateral</a>: the diagonals are of equal length.</li> <li>Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram.</li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential quadrilateral</a>: the four extensions of the sides are tangent to an <a href="/wiki/Excircle" class="mw-redirect" title="Excircle">excircle</a>.</li> <li>An equilic quadrilateral has two opposite equal sides that when extended, meet at 60°.</li> <li>A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length.<a href="#cite_note-6">[6]</a></li> <li>A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.<a href="#cite_note-7">[7]</a></li> <li>A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.<a href="#cite_note-8">[8]</a></li> <li>A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.<a href="#cite_note-9">[9]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Concave_quadrilaterals">Concave quadrilaterals</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=3" title="Edit section: Concave quadrilaterals">edit</a>]</div> In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral. <ul><li>A dart (or arrowhead) is a <a href="/wiki/Concave_polygon" title="Concave polygon">concave</a> quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Complex_quadrilaterals">Complex quadrilaterals</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=4" title="Edit section: Complex quadrilaterals">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:DU21_facets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/DU21_facets.png/180px-DU21_facets.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/DU21_facets.png/270px-DU21_facets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/DU21_facets.png/360px-DU21_facets.png 2x" data-file-width="1200" data-file-height="1200" /></a><figcaption>An antiparallelogram</figcaption></figure> A <a href="/wiki/List_of_self-intersecting_polygons" title="List of self-intersecting polygons">self-intersecting</a> quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, <a href="/wiki/Butterfly" title="Butterfly">butterfly</a> quadrilateral or <a href="/wiki/Bow-tie" class="mw-redirect" title="Bow-tie">bow-tie</a> quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two <a href="/wiki/Acute_angle" class="mw-redirect" title="Acute angle">acute</a> and two <a href="/wiki/Reflex_angle" class="mw-redirect" title="Reflex angle">reflex</a>, all on the left or all on the right as the figure is traced out) add up to 720°.<a href="#cite_note-10">[10]</a> <ul><li><a href="/wiki/Isosceles_trapezoid#Self-intersections" title="Isosceles trapezoid">Crossed trapezoid</a> (US) or trapezium (Commonwealth):<a href="#cite_note-11">[11]</a> a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a>).</li> <li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a>: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>).</li> <li><a href="/wiki/Crossed_rectangle" class="mw-redirect" title="Crossed rectangle">Crossed rectangle</a>: an antiparallelogram whose sides are two opposite sides and the two diagonals of a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>, hence having one pair of parallel opposite sides.</li> <li><a href="/wiki/Square#Crossed_square" title="Square">Crossed square</a>: a special case of a crossed rectangle where two of the sides intersect at right angles.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Special_line_segments">Special line segments</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=5" title="Edit section: Special line segments">edit</a>]</div> The two <a href="/wiki/Diagonal" title="Diagonal">diagonals</a> of a convex quadrilateral are the <a href="/wiki/Line_segment" title="Line segment">line segments</a> that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides.<a href="#cite_note-12">[12]</a> They intersect at the "vertex centroid" of the quadrilateral (see <a href="#Remarkable_points_and_lines_in_a_convex_quadrilateral">§ Remarkable points and lines in a convex quadrilateral</a> below). The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.<a href="#cite_note-13">[13]</a> <div class="mw-heading mw-heading2"><h2 id="Area_of_a_convex_quadrilateral">Area of a convex quadrilateral</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=6" title="Edit section: Area of a convex quadrilateral">edit</a>]</div> There are various general formulas for the <a href="/wiki/Area" title="Area">area</a> K of a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD and d = DA. <div class="mw-heading mw-heading3"><h3 id="Trigonometric_formulas">Trigonometric formulas</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=7" title="Edit section: Trigonometric formulas">edit</a>]</div> The area can be expressed in trigonometric terms as<a href="#cite_note-:1-14">[14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}pq\sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>p</mi> <mi>q</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}pq\sin \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb422f3244c36997bed91a3574fe0db7ce0b8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.429ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}pq\sin \theta ,}"></dd></dl> where the lengths of the diagonals are p and q and the angle between them is θ.<a href="#cite_note-15">[15]</a> In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {pq}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>p</mi> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {pq}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1650f1d3797c628d94e5a092ff3851b730c979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.584ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {pq}{2}}}"> since θ is 90°. The area can be also expressed in terms of bimedians as<a href="#cite_note-Josefsson4-16">[16]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=mn\sin \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>m</mi> <mi>n</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=mn\sin \varphi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1974d756d5a624225461818948e149c577ad29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.396ex; height:2.676ex;" alt="{\displaystyle K=mn\sin \varphi ,}"></dd></dl> where the lengths of the bimedians are m and n and the angle between them is φ. <a href="/wiki/Bretschneider%27s_formula" title="Bretschneider's formula">Bretschneider's formula</a><a href="#cite_note-17">[17]</a><a href="#cite_note-:1-14">[14]</a> expresses the area in terms of the sides and two opposite angles: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;[1+\cos(A+C)]}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\,\cos ^{2}{\tfrac {1}{2}}(A+C)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mspace width="thinmathspace" /> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;[1+\cos(A+C)]}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\,\cos ^{2}{\tfrac {1}{2}}(A+C)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6195867b271ce1ef16fc42e8371f1fcf1657f0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:61.811ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;[1+\cos(A+C)]}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\,\cos ^{2}{\tfrac {1}{2}}(A+C)}}\end{aligned}}}"></dd></dl> where the sides in sequence are a, b, c, d, where s is the semiperimeter, and A and C are two (in fact, any two) opposite angles. This reduces to <a href="/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a> for the area of a cyclic quadrilateral—when A + C = 180° . Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}ad\sin {A}+{\tfrac {1}{2}}bc\sin {C}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>d</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>b</mi> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}ad\sin {A}+{\tfrac {1}{2}}bc\sin {C}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a70bf585f6eb899d823ab9c028808f95243196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.187ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}ad\sin {A}+{\tfrac {1}{2}}bc\sin {C}.}"></dd></dl> In the case of a cyclic quadrilateral, the latter formula becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}(ad+bc)\sin {A}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}(ad+bc)\sin {A}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88698617b7222ea84fb512999f74d1f7bd09086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.942ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}(ad+bc)\sin {A}.}"> In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=ab\cdot \sin {A}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=ab\cdot \sin {A}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70722d5cac90f9bbfbbff084058a295f8db48638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.704ex; height:2.176ex;" alt="{\displaystyle K=ab\cdot \sin {A}.}"> Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, as long θ is not 90°:<a href="#cite_note-Mitchell-18">[18]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{4}}\left|\tan \theta \right|\cdot \left|a^{2}+c^{2}-b^{2}-d^{2}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>|</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{4}}\left|\tan \theta \right|\cdot \left|a^{2}+c^{2}-b^{2}-d^{2}\right|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7a981cb95357bd5d2af46dd35cf8d616439723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.538ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{4}}\left|\tan \theta \right|\cdot \left|a^{2}+c^{2}-b^{2}-d^{2}\right|.}"></dd></dl> In the case of a parallelogram, the latter formula becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}\left|\tan \theta \right|\cdot \left|a^{2}-b^{2}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>|</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}\left|\tan \theta \right|\cdot \left|a^{2}-b^{2}\right|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a83bc9f432b80bfc1e7690bdaf3bb525ddbd0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.523ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}\left|\tan \theta \right|\cdot \left|a^{2}-b^{2}\right|.}"> Another area formula including the sides a, b, c, d is<a href="#cite_note-Josefsson4-16">[16]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}{\sqrt {{\bigl (}(a^{2}+c^{2})-2x^{2}{\bigr )}{\bigl (}(b^{2}+d^{2})-2x^{2}{\bigr )}}}\sin {\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}{\sqrt {{\bigl (}(a^{2}+c^{2})-2x^{2}{\bigr )}{\bigl (}(b^{2}+d^{2})-2x^{2}{\bigr )}}}\sin {\varphi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de13c339159849d2703d5675a63b2a767c045e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.298ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{2}}{\sqrt {{\bigl (}(a^{2}+c^{2})-2x^{2}{\bigr )}{\bigl (}(b^{2}+d^{2})-2x^{2}{\bigr )}}}\sin {\varphi }}"></dd></dl> where x is the distance between the midpoints of the diagonals, and φ is the angle between the <a class="mw-selflink-fragment" href="#Special_line_segments">bimedians</a>. The last trigonometric area formula including the sides a, b, c, d and the angle α (between a and b) is:<a href="#cite_note-19">[19]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}ab\sin {\alpha }+{\tfrac {1}{4}}{\sqrt {4c^{2}d^{2}-(c^{2}+d^{2}-a^{2}-b^{2}+2ab\cos {\alpha })^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}ab\sin {\alpha }+{\tfrac {1}{4}}{\sqrt {4c^{2}d^{2}-(c^{2}+d^{2}-a^{2}-b^{2}+2ab\cos {\alpha })^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af62c77b929550016c0e5f2172236254b38ab1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:61.63ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{2}}ab\sin {\alpha }+{\tfrac {1}{4}}{\sqrt {4c^{2}d^{2}-(c^{2}+d^{2}-a^{2}-b^{2}+2ab\cos {\alpha })^{2}}},}"></dd></dl> which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle α), by just changing the first sign + to -. <div class="mw-heading mw-heading3"><h3 id="Non-trigonometric_formulas">Non-trigonometric formulas</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=8" title="Edit section: Non-trigonometric formulas">edit</a>]</div> The following two formulas express the area in terms of the sides a, b, c and d, the <a href="/wiki/Semiperimeter#Quadrilaterals" title="Semiperimeter">semiperimeter</a> s, and the diagonals p, q: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+pq)(ac+bd-pq)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mi>p</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+pq)(ac+bd-pq)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcbb01d08917fc3edbaf23e36a08ba376462043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.402ex; height:4.843ex;" alt="{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+pq)(ac+bd-pq)}},}"> <a href="#cite_note-20">[20]</a></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{4}}{\sqrt {4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{4}}{\sqrt {4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a497c04c22e1ad983d44c4a1444520a2bda67610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:38.528ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{4}}{\sqrt {4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}}.}"> <a href="#cite_note-21">[21]</a></dd></dl> The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then pq = ac + bd. The area can also be expressed in terms of the bimedians m, n and the diagonals p, q: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}{\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}{\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6160ff5c0893c088302ca5e1ed1ed6dbc6234b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.972ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{2}}{\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}},}"> <a href="#cite_note-22">[22]</a></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}{\sqrt {p^{2}q^{2}-(m^{2}-n^{2})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}{\sqrt {p^{2}q^{2}-(m^{2}-n^{2})^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2dc8282ed3d6e1fd84870cc3d34eecebc358d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:28.238ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{2}}{\sqrt {p^{2}q^{2}-(m^{2}-n^{2})^{2}}}.}"> <a href="#cite_note-Josefsson3-23">[23]</a>: Thm. 7 </dd></dl> In fact, any three of the four values m, n, p, and q suffice for determination of the area, since in any quadrilateral the four values are related by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ffcfca88c30dfa5f35dd88ce8c062d62bf367e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:22.388ex; height:3.176ex;" alt="{\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}"><a href="#cite_note-Altshiller-Court-24">[24]</a>: p. 126  The corresponding expressions are:<a href="#cite_note-Josefsson6-25">[25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}{\sqrt {[(m+n)^{2}-p^{2}]\cdot [p^{2}-(m-n)^{2}]}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}{\sqrt {[(m+n)^{2}-p^{2}]\cdot [p^{2}-(m-n)^{2}]}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e6f895931ea57ab71ca3ec4d03175e3d73bef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:42.466ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{2}}{\sqrt {[(m+n)^{2}-p^{2}]\cdot [p^{2}-(m-n)^{2}]}},}"></dd></dl> if the lengths of two bimedians and one diagonal are given, and<a href="#cite_note-Josefsson6-25">[25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{4}}{\sqrt {[(p+q)^{2}-4m^{2}]\cdot [4m^{2}-(p-q)^{2}]}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mn>4</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{4}}{\sqrt {[(p+q)^{2}-4m^{2}]\cdot [4m^{2}-(p-q)^{2}]}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2f523551c427056fa7199b6ad0d093b770d4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:44.14ex; height:4.843ex;" alt="{\displaystyle K={\tfrac {1}{4}}{\sqrt {[(p+q)^{2}-4m^{2}]\cdot [4m^{2}-(p-q)^{2}]}},}"></dd></dl> if the lengths of two diagonals and one bimedian are given. <div class="mw-heading mw-heading3"><h3 id="Vector_formulas">Vector formulas</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=9" title="Edit section: Vector formulas">edit</a>]</div> The area of a quadrilateral ABCD can be calculated using <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a>. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}|\mathbf {AC} \times \mathbf {BD} |,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}|\mathbf {AC} \times \mathbf {BD} |,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e48a73a24dc41afdcd2b345c0ae2a5b0b6489e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.505ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}|\mathbf {AC} \times \mathbf {BD} |,}"></dd></dl> which is half the magnitude of the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of vectors AC and BD. In two-dimensional Euclidean space, expressing vector AC as a <a href="/wiki/Euclidean_vector#In_Cartesian_space" title="Euclidean vector">free vector in Cartesian space</a> equal to (x1,y1) and BD as (x2,y2), this can be rewritten as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}|x_{1}y_{2}-x_{2}y_{1}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}|x_{1}y_{2}-x_{2}y_{1}|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8e7ee38a0432822b333b787a087e726e854947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.758ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}|x_{1}y_{2}-x_{2}y_{1}|.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Diagonals">Diagonals</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=10" title="Edit section: Diagonals">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Properties_of_the_diagonals_in_quadrilaterals">Properties of the diagonals in quadrilaterals</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=11" title="Edit section: Properties of the diagonals in quadrilaterals">edit</a>]</div> In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>, and if their diagonals have equal length.<a href="#cite_note-26">[26]</a> The list applies to the most general cases, and excludes named subsets. <table class="wikitable"> <tbody><tr> <th scope="col">Quadrilateral </th> <th scope="col">Bisecting diagonals </th> <th scope="col">Perpendicular diagonals </th> <th scope="col">Equal diagonals </th></tr> <tr> <th scope="row"><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td>See note 1</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th scope="row"><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td>See note 1</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th scope="row"><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th scope="row"><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a> </th> <td>See note 2</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td>See note 2 </td></tr> <tr> <th scope="row"><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th scope="row"><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th scope="row"><a href="/wiki/Square" title="Square">Square</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr></tbody></table> <ul><li>Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral.</li> <li>Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Lengths_of_the_diagonals">Lengths of the diagonals</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=12" title="Edit section: Lengths of the diagonals">edit</a>]</div> The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a> on each triangle formed by one diagonal and two sides of the quadrilateral. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\sqrt {a^{2}+b^{2}-2ab\cos {B}}}={\sqrt {c^{2}+d^{2}-2cd\cos {D}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>c</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\sqrt {a^{2}+b^{2}-2ab\cos {B}}}={\sqrt {c^{2}+d^{2}-2cd\cos {D}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8ad154c5671d1785ffc254cd237f8c331b30aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:50.368ex; height:3.509ex;" alt="{\displaystyle p={\sqrt {a^{2}+b^{2}-2ab\cos {B}}}={\sqrt {c^{2}+d^{2}-2cd\cos {D}}}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\sqrt {a^{2}+d^{2}-2ad\cos {A}}}={\sqrt {b^{2}+c^{2}-2bc\cos {C}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\sqrt {a^{2}+d^{2}-2ad\cos {A}}}={\sqrt {b^{2}+c^{2}-2bc\cos {C}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d73a44cb88f98d1f324621ca5ec14e40bb60c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.647ex; height:3.509ex;" alt="{\displaystyle q={\sqrt {a^{2}+d^{2}-2ad\cos {A}}}={\sqrt {b^{2}+c^{2}-2bc\cos {C}}}.}"></dd></dl> Other, more symmetric formulas for the lengths of the diagonals, are<a href="#cite_note-27">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\sqrt {\frac {(ac+bd)(ad+bc)-2abcd(\cos {B}+\cos {D})}{ab+cd}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\sqrt {\frac {(ac+bd)(ad+bc)-2abcd(\cos {B}+\cos {D})}{ab+cd}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178dfdb70d1e5e3a7f7f3d0614d319e0c68dc15b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:49.504ex; height:7.676ex;" alt="{\displaystyle p={\sqrt {\frac {(ac+bd)(ad+bc)-2abcd(\cos {B}+\cos {D})}{ab+cd}}}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\sqrt {\frac {(ab+cd)(ac+bd)-2abcd(\cos {A}+\cos {C})}{ad+bc}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\sqrt {\frac {(ab+cd)(ac+bd)-2abcd(\cos {A}+\cos {C})}{ad+bc}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb0d3565d72fc0ed6477b0f51f993dbec4c2c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.783ex; height:7.676ex;" alt="{\displaystyle q={\sqrt {\frac {(ab+cd)(ac+bd)-2abcd(\cos {A}+\cos {C})}{ad+bc}}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generalizations_of_the_parallelogram_law_and_Ptolemy's_theorem">Generalizations of the parallelogram law and Ptolemy's theorem</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=13" title="Edit section: Generalizations of the parallelogram law and Ptolemy's theorem">edit</a>]</div> In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d207f329643a75a3c3c856d040896474e262a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.873ex; height:3.009ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}}"></dd></dl> where x is the distance between the midpoints of the diagonals.<a href="#cite_note-Altshiller-Court-24">[24]</a>: p.126  This is sometimes known as <a href="/wiki/Euler%27s_quadrilateral_theorem" title="Euler's quadrilateral theorem">Euler's quadrilateral theorem</a> and is a generalization of the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a>. The German mathematician <a href="/wiki/Carl_Anton_Bretschneider" title="Carl Anton Bretschneider">Carl Anton Bretschneider</a> derived in 1842 the following generalization of <a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a>, regarding the product of the diagonals in a convex quadrilateral<a href="#cite_note-28">[28]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos {(A+C)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos {(A+C)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae09985fb3f93afa18f4c32a5a0fcc36ba072351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:40.199ex; height:3.176ex;" alt="{\displaystyle p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos {(A+C)}.}"></dd></dl> This relation can be considered to be a <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a> for a quadrilateral. In a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>, where A + C = 180°, it reduces to pq = ac + bd. Since cos (A + C) ≥ −1, it also gives a proof of Ptolemy's inequality. <div class="mw-heading mw-heading3"><h3 id="Other_metric_relations">Other metric relations</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=14" title="Edit section: Other metric relations">edit</a>]</div> If X and Y are the feet of the normals from B and D to the diagonal AC = p in a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, then<a href="#cite_note-Josefsson-29">[29]</a>: p.14  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XY={\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XY={\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f8f015a4360feb7a3496e3ea02441b6fdf8bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.819ex; height:6.343ex;" alt="{\displaystyle XY={\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.}"></dd></dl> In a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, and where the diagonals intersect at E, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>f</mi> <mi>g</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>g</mi> <mi>h</mi> <mo>+</mo> <mi>c</mi> <mi>e</mi> <mi>f</mi> <mo>+</mo> <mi>b</mi> <mi>e</mi> <mi>h</mi> <mo>+</mo> <mi>d</mi> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>g</mi> <mi>h</mi> <mo>+</mo> <mi>c</mi> <mi>e</mi> <mi>f</mi> <mo>−</mo> <mi>b</mi> <mi>e</mi> <mi>h</mi> <mo>−</mo> <mi>d</mi> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80e0679eab13c6c23d339e4229a1e94ace1e343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:86.306ex; height:2.843ex;" alt="{\displaystyle efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)}"></dd></dl> where e = AE, f = BE, g = CE, and h = DE.<a href="#cite_note-30">[30]</a> The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p, q and the four side lengths a, b, c, d of a quadrilateral are related<a href="#cite_note-:1-14">[14]</a> by the <a href="/wiki/Distance_geometry#Cayley.E2.80.93Menger_determinants" title="Distance geometry">Cayley-Menger</a> <a href="/wiki/Determinant" title="Determinant">determinant</a>, as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91d61c86c38ecf5b26a0a0f66d4da8224975de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.266ex; width:31.893ex; height:16.176ex;" alt="{\displaystyle \det {\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Angle_bisectors">Angle bisectors</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=15" title="Edit section: Angle bisectors">edit</a>]</div> The internal <a href="/wiki/Angle_bisector" class="mw-redirect" title="Angle bisector">angle bisectors</a> of a convex quadrilateral either form a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a><a href="#cite_note-Altshiller-Court-24">[24]</a>: p.127  (that is, the four intersection points of adjacent angle bisectors are <a href="/wiki/Concyclic_points" title="Concyclic points">concyclic</a>) or they are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a>. In the latter case the quadrilateral is a <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a>. In quadrilateral ABCD, if the <a href="/wiki/Bisection#Of_angles" title="Bisection">angle bisectors</a> of A and C meet on diagonal BD, then the angle bisectors of B and D meet on diagonal AC.<a href="#cite_note-31">[31]</a> <div class="mw-heading mw-heading2"><h2 id="Bimedians">Bimedians</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=16" title="Edit section: Bimedians">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Varignon%27s_theorem" title="Varignon's theorem">Varignon's theorem</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Varignon_theorem_convex.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Varignon_theorem_convex.png/300px-Varignon_theorem_convex.png" decoding="async" width="300" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Varignon_theorem_convex.png/450px-Varignon_theorem_convex.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Varignon_theorem_convex.png/600px-Varignon_theorem_convex.png 2x" data-file-width="706" data-file-height="477" /></a><figcaption>The Varignon parallelogram EFGH</figcaption></figure> The <a class="mw-selflink-fragment" href="#Special_line_segments">bimedians</a> of a quadrilateral are the line segments connecting the <a href="/wiki/Midpoint" title="Midpoint">midpoints</a> of the opposite sides. The intersection of the bimedians is the <a href="/wiki/Centroid" title="Centroid">centroid</a> of the vertices of the quadrilateral.<a href="#cite_note-:1-14">[14]</a> The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> called the <a href="/wiki/Varignon%27s_theorem" title="Varignon's theorem">Varignon parallelogram</a>. It has the following properties: <ul><li>Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.</li> <li>A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.</li> <li>The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.<a href="#cite_note-32">[32]</a></li> <li>The <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.</li> <li>The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.</li></ul> The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> and are all bisected by their point of intersection.<a href="#cite_note-Altshiller-Court-24">[24]</a>: p.125  In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1cb9942329f76eb2454b706ec6c63ce8337272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:38.157ex; height:4.843ex;" alt="{\displaystyle m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}}"></dd></dl> where p and q are the length of the diagonals.<a href="#cite_note-33">[33]</a> The length of the bimedian that connects the midpoints of the sides b and d is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71beb3c187aab28cf32df989830aff9041280d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:36.35ex; height:4.843ex;" alt="{\displaystyle n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.}"></dd></dl> Hence<a href="#cite_note-Altshiller-Court-24">[24]</a>: p.126  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0b98b4334dd6400016e189f7a1d151ce92c1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:22.388ex; height:3.176ex;" alt="{\displaystyle \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}"></dd></dl> This is also a <a href="/wiki/Corollary" title="Corollary">corollary</a> to the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a> applied in the Varignon parallelogram. The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<a href="#cite_note-Josefsson3-23">[23]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9903d87f5dc01b783a55324c992bfe6771ff37d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.644ex; height:4.843ex;" alt="{\displaystyle m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e358f396a3695ce1f75f9f1c10cc5119b9d864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.666ex; height:4.843ex;" alt="{\displaystyle n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.}"></dd></dl> Note that the two opposite sides in these formulas are not the two that the bimedian connects. In a convex quadrilateral, there is the following <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dual</a> connection between the bimedians and the diagonals:<a href="#cite_note-Josefsson-29">[29]</a> <ul><li>The two bimedians have equal length <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the two diagonals are <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>.</li> <li>The two bimedians are perpendicular if and only if the two diagonals have equal length.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Trigonometric_identities">Trigonometric identities</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=17" title="Edit section: Trigonometric identities">edit</a>]</div> The four angles of a simple quadrilateral ABCD satisfy the following identities:<a href="#cite_note-34">[34]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin A+\sin B+\sin C+\sin D=4\sin {\tfrac {1}{2}}(A+B)\,\sin {\tfrac {1}{2}}(A+C)\,\sin {\tfrac {1}{2}}(A+D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>D</mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin A+\sin B+\sin C+\sin D=4\sin {\tfrac {1}{2}}(A+B)\,\sin {\tfrac {1}{2}}(A+C)\,\sin {\tfrac {1}{2}}(A+D)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27630643f063668b25a1fb7515e538566129ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:74.222ex; height:3.509ex;" alt="{\displaystyle \sin A+\sin B+\sin C+\sin D=4\sin {\tfrac {1}{2}}(A+B)\,\sin {\tfrac {1}{2}}(A+C)\,\sin {\tfrac {1}{2}}(A+D)}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tan A\,\tan {B}-\tan C\,\tan D}{\tan A\,\tan C-\tan B\,\tan D}}={\frac {\tan(A+C)}{\tan(A+B)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>C</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>D</mi> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>C</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>B</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tan A\,\tan {B}-\tan C\,\tan D}{\tan A\,\tan C-\tan B\,\tan D}}={\frac {\tan(A+C)}{\tan(A+B)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a89b505db538675a76ae78ccbca4b7f2b4722b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.51ex; height:6.509ex;" alt="{\displaystyle {\frac {\tan A\,\tan {B}-\tan C\,\tan D}{\tan A\,\tan C-\tan B\,\tan D}}={\frac {\tan(A+C)}{\tan(A+B)}}.}"></dd></dl> Also,<a href="#cite_note-35">[35]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tan A+\tan B+\tan C+\tan D}{\cot A+\cot B+\cot C+\cot D}}=\tan {A}\tan {B}\tan {C}\tan {D}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>B</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>C</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>D</mi> </mrow> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>A</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>B</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>C</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tan A+\tan B+\tan C+\tan D}{\cot A+\cot B+\cot C+\cot D}}=\tan {A}\tan {B}\tan {C}\tan {D}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b43d8cf48ab73bc7dfb792e7b85077a7b31e179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:58.633ex; height:5.676ex;" alt="{\displaystyle {\frac {\tan A+\tan B+\tan C+\tan D}{\cot A+\cot B+\cot C+\cot D}}=\tan {A}\tan {B}\tan {C}\tan {D}.}"></dd></dl> In the last two formulas, no angle is allowed to be a <a href="/wiki/Right_angle" title="Right angle">right angle</a>, since tan 90° is not defined. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> be the sides of a convex quadrilateral, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is the semiperimeter, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> are opposite angles, then<a href="#cite_note-36">[36]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ad\sin ^{2}{\tfrac {1}{2}}A+bc\cos ^{2}{\tfrac {1}{2}}C=(s-a)(s-d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>d</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>A</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ad\sin ^{2}{\tfrac {1}{2}}A+bc\cos ^{2}{\tfrac {1}{2}}C=(s-a)(s-d)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427ac4d0dfa0929e8adaf936d0d599f0f6a2e798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.764ex; height:3.509ex;" alt="{\displaystyle ad\sin ^{2}{\tfrac {1}{2}}A+bc\cos ^{2}{\tfrac {1}{2}}C=(s-a)(s-d)}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bc\sin ^{2}{\tfrac {1}{2}}C+ad\cos ^{2}{\tfrac {1}{2}}A=(s-b)(s-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>c</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>C</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bc\sin ^{2}{\tfrac {1}{2}}C+ad\cos ^{2}{\tfrac {1}{2}}A=(s-b)(s-c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d19cda07dbbaebffda7e1ca6b4902713bcce20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.323ex; height:3.509ex;" alt="{\displaystyle bc\sin ^{2}{\tfrac {1}{2}}C+ad\cos ^{2}{\tfrac {1}{2}}A=(s-b)(s-c)}">.</dd></dl> We can use these identities to derive the <a href="/wiki/Bretschneider%27s_Formula" class="mw-redirect" title="Bretschneider's Formula">Bretschneider's Formula</a>. <div class="mw-heading mw-heading2"><h2 id="Inequalities">Inequalities</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=18" title="Edit section: Inequalities">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Area">Area</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=19" title="Edit section: Area">edit</a>]</div> If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies<a href="#cite_note-37">[37]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{4}}(a+c)(b+d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{4}}(a+c)(b+d)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236dd3b6f77de775b2cbd4954dd23ba90b56ee44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.572ex; height:3.509ex;" alt="{\displaystyle K\leq {\tfrac {1}{4}}(a+c)(b+d)}"> with equality only for a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1341d919b6e494479203d3503e26fb0c3b2f5b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.822ex; height:3.509ex;" alt="{\displaystyle K\leq {\tfrac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2})}"> with equality only for a <a href="/wiki/Square" title="Square">square</a>.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{4}}(p^{2}+q^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <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> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{4}}(p^{2}+q^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba62309c39f6ab9bd3f08803b86852d6cf7cebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.83ex; height:3.509ex;" alt="{\displaystyle K\leq {\tfrac {1}{4}}(p^{2}+q^{2})}"> with equality only if the diagonals are perpendicular and equal.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff68993009bd9c9c7d69c33e5ae3010e81c6c26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:27.115ex; height:4.843ex;" alt="{\displaystyle K\leq {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}}"> with equality only for a rectangle.<a href="#cite_note-Josefsson4-16">[16]</a></dd></dl> From <a href="/wiki/Bretschneider%27s_formula" title="Bretschneider's formula">Bretschneider's formula</a> it directly follows that the area of a quadrilateral satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\sqrt {(s-a)(s-b)(s-c)(s-d)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\sqrt {(s-a)(s-b)(s-c)(s-d)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b41e01c7b9154659897b6d0f6747d304bd2d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.899ex; height:4.843ex;" alt="{\displaystyle K\leq {\sqrt {(s-a)(s-b)(s-c)(s-d)}}}"></dd></dl> with equality <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the quadrilateral is <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic</a> or degenerate such that one side is equal to the sum of the other three (it has collapsed into a <a href="/wiki/Line_segment" title="Line segment">line segment</a>, so the area is zero). Also, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\sqrt {abcd}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\sqrt {abcd}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32254af3f173cbd25c87defc4abb18f015b89f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.197ex; height:3.009ex;" alt="{\displaystyle K\leq {\sqrt {abcd}},}"></dd></dl> with equality for a <a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">bicentric quadrilateral</a> or a rectangle. The area of any quadrilateral also satisfies the inequality<a href="#cite_note-Alsina-38">[38]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle K\leq {\tfrac {1}{2}}{\sqrt[{3}]{(ab+cd)(ac+bd)(ad+bc)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle K\leq {\tfrac {1}{2}}{\sqrt[{3}]{(ab+cd)(ac+bd)(ad+bc)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423a43c2641c34f91f0be6e9ccf2ba7c9cc8412a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.092ex; height:4.843ex;" alt="{\displaystyle \displaystyle K\leq {\tfrac {1}{2}}{\sqrt[{3}]{(ab+cd)(ac+bd)(ad+bc)}}.}"></dd></dl> Denoting the perimeter as L, we have<a href="#cite_note-Alsina-38">[38]</a>: p.114  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{16}}L^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mstyle> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{16}}L^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e83e88d301f1516f5d97522c3a288ebef18d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.928ex; height:3.676ex;" alt="{\displaystyle K\leq {\tfrac {1}{16}}L^{2},}"></dd></dl> with equality only in the case of a square. The area of a convex quadrilateral also satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{2}}pq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{2}}pq}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd53be28f2cdd679dfb0e3c837034646c597bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.061ex; height:3.509ex;" alt="{\displaystyle K\leq {\tfrac {1}{2}}pq}"></dd></dl> for diagonal lengths p and q, with equality if and only if the diagonals are perpendicular. Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K and diagonals AC = p, BD = q. Then<a href="#cite_note-39">[39]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{8}}(a^{2}+b^{2}+c^{2}+d^{2}+p^{2}+q^{2}+pq-ac-bd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>q</mi> <mo>−</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{8}}(a^{2}+b^{2}+c^{2}+d^{2}+p^{2}+q^{2}+pq-ac-bd)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7bf03c6f257720d60c4131fcc80b9c8d6573960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:51.071ex; height:3.676ex;" alt="{\displaystyle K\leq {\tfrac {1}{8}}(a^{2}+b^{2}+c^{2}+d^{2}+p^{2}+q^{2}+pq-ac-bd)}"> with equality only for a square.</dd></dl> Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K, then the following inequality holds:<a href="#cite_note-40">[40]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\frac {1}{3+{\sqrt {3}}}}(ab+ac+ad+bc+bd+cd)-{\frac {1}{2(1+{\sqrt {3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\frac {1}{3+{\sqrt {3}}}}(ab+ac+ad+bc+bd+cd)-{\frac {1}{2(1+{\sqrt {3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/642ae81c84d51d098cc75490e2de54995300f062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.267ex; height:6.343ex;" alt="{\displaystyle K\leq {\frac {1}{3+{\sqrt {3}}}}(ab+ac+ad+bc+bd+cd)-{\frac {1}{2(1+{\sqrt {3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})}"> with equality only for a square.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Diagonals_and_bimedians">Diagonals and bimedians</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=20" title="Edit section: Diagonals and bimedians">edit</a>]</div> A corollary to Euler's quadrilateral theorem is the inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61351bf78f29335853987f542ac945099e6d592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.487ex; height:3.009ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}}"></dd></dl> where equality holds if and only if the quadrilateral is a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> also generalized <a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a>, which is an equality in a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>, into an inequality for a convex quadrilateral. It states that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pq\leq ac+bd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>q</mi> <mo>≤</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pq\leq ac+bd}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270efed7debd13222f4c73bad75713339edb9476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.717ex; height:2.509ex;" alt="{\displaystyle pq\leq ac+bd}"></dd></dl> where there is equality <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the quadrilateral is cyclic.<a href="#cite_note-Altshiller-Court-24">[24]</a>: p.128–129  This is often called <a href="/wiki/Ptolemy%27s_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a>. In any convex quadrilateral the bimedians m, n and the diagonals p, q are related by the inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pq\leq m^{2}+n^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>q</mi> <mo>≤</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pq\leq m^{2}+n^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09454e09ddb273228ede155d9e71a75a40b961ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.458ex; height:3.009ex;" alt="{\displaystyle pq\leq m^{2}+n^{2},}"></dd></dl> with equality holding if and only if the diagonals are equal.<a href="#cite_note-J2014-41">[41]</a>: Prop.1  This follows directly from the quadrilateral identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{2}+n^{2}={\tfrac {1}{2}}(p^{2}+q^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <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> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}+n^{2}={\tfrac {1}{2}}(p^{2}+q^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bd9595f453af70facb753abb46185aeefcd871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.794ex; height:3.509ex;" alt="{\displaystyle m^{2}+n^{2}={\tfrac {1}{2}}(p^{2}+q^{2}).}"> <div class="mw-heading mw-heading3"><h3 id="Sides">Sides</h3>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=21" title="Edit section: Sides">edit</a>]</div> The sides a, b, c, and d of any quadrilateral satisfy<a href="#cite_note-Crux-42">[42]</a>: p.228, #275  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}+c^{2}>{\tfrac {1}{3}}d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}>{\tfrac {1}{3}}d^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127d03b77eb6d407c28863396447be7dd1c85e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.107ex; height:3.676ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}>{\tfrac {1}{3}}d^{2}}"></dd></dl> and<a href="#cite_note-Crux-42">[42]</a>: p.234, #466  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{4}+b^{4}+c^{4}\geq {\tfrac {1}{27}}d^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>27</mn> </mfrac> </mstyle> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{4}+b^{4}+c^{4}\geq {\tfrac {1}{27}}d^{4}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49ce333c1567b7630551b732f460a6f39913a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.575ex; height:3.676ex;" alt="{\displaystyle a^{4}+b^{4}+c^{4}\geq {\tfrac {1}{27}}d^{4}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Maximum_and_minimum_properties">Maximum and minimum properties</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=22" title="Edit section: Maximum and minimum properties">edit</a>]</div> Among all quadrilaterals with a given <a href="/wiki/Perimeter" title="Perimeter">perimeter</a>, the one with the largest area is the <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a>. This is called the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric theorem</a> for quadrilaterals. It is a direct consequence of the area inequality<a href="#cite_note-Alsina-38">[38]</a>: p.114  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\leq {\tfrac {1}{16}}L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mstyle> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\leq {\tfrac {1}{16}}L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6587904ece9061f95aaded3a026ac50ca8949edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.282ex; height:3.676ex;" alt="{\displaystyle K\leq {\tfrac {1}{16}}L^{2}}"></dd></dl> where K is the area of a convex quadrilateral with perimeter L. Equality holds <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter. The quadrilateral with given side lengths that has the <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximum</a> area is the <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>.<a href="#cite_note-Peter-43">[43]</a> Of all convex quadrilaterals with given diagonals, the <a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">orthodiagonal quadrilateral</a> has the largest area.<a href="#cite_note-Alsina-38">[38]</a>: p.119  This is a direct consequence of the fact that the area of a convex quadrilateral satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}pq\sin {\theta }\leq {\tfrac {1}{2}}pq,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>p</mi> <mi>q</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>p</mi> <mi>q</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}pq\sin {\theta }\leq {\tfrac {1}{2}}pq,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5198c0272ce42ab760d0fdd72ec64d9c84abbadf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.424ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}pq\sin {\theta }\leq {\tfrac {1}{2}}pq,}"></dd></dl> where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°. If P is an interior point in a convex quadrilateral ABCD, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP+BP+CP+DP\geq AC+BD.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>P</mi> <mo>+</mo> <mi>B</mi> <mi>P</mi> <mo>+</mo> <mi>C</mi> <mi>P</mi> <mo>+</mo> <mi>D</mi> <mi>P</mi> <mo>≥</mo> <mi>A</mi> <mi>C</mi> <mo>+</mo> <mi>B</mi> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AP+BP+CP+DP\geq AC+BD.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52215f60c8de53fd7e51f58610f47efde1fa7b72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.484ex; height:2.343ex;" alt="{\displaystyle AP+BP+CP+DP\geq AC+BD.}"></dd></dl> From this inequality it follows that the point inside a quadrilateral that <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">minimizes</a> the sum of distances to the <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> is the intersection of the diagonals. Hence that point is the <a href="/wiki/Fermat_point" title="Fermat point">Fermat point</a> of a convex quadrilateral.<a href="#cite_note-autogenerated1-44">[44]</a>: p.120  <div class="mw-heading mw-heading2"><h2 id="Remarkable_points_and_lines_in_a_convex_quadrilateral">Remarkable points and lines in a convex quadrilateral</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=23" title="Edit section: Remarkable points and lines in a convex quadrilateral">edit</a>]</div> The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just <a href="/wiki/Centroid" title="Centroid">centroid</a> (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.<a href="#cite_note-45">[45]</a> The "vertex centroid" is the intersection of the two <a class="mw-selflink-fragment" href="#Special_line_segments">bimedians</a>.<a href="#cite_note-46">[46]</a> As with any polygon, the x and y coordinates of the vertex centroid are the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic means</a> of the x and y coordinates of the vertices. The "area centroid" of quadrilateral ABCD can be constructed in the following way. Let Ga, Gb, Gc, Gd be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines GaGc and GbGd.<a href="#cite_note-Myakishev-47">[47]</a> In a general convex quadrilateral ABCD, there are no natural analogies to the <a href="/wiki/Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a> and <a href="/wiki/Orthocenter" title="Orthocenter">orthocenter</a> of a <a href="/wiki/Triangle" title="Triangle">triangle</a>. But two such points can be constructed in the following way. Let Oa, Ob, Oc, Od be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by Ha, Hb, Hc, Hd the orthocenters in the same triangles. Then the intersection of the lines OaOc and ObOd is called the <a href="/wiki/Circumcenter_of_mass" title="Circumcenter of mass">quasicircumcenter</a>, and the intersection of the lines HaHc and HbHd is called the quasiorthocenter of the convex quadrilateral.<a href="#cite_note-Myakishev-47">[47]</a> These points can be used to define an <a href="/wiki/Euler_line" title="Euler line">Euler line</a> of a quadrilateral. In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a> in this order, and HG = 2GO.<a href="#cite_note-Myakishev-47">[47]</a> There can also be defined a quasinine-point center E as the intersection of the lines EaEc and EbEd, where Ea, Eb, Ec, Ed are the <a href="/wiki/Nine-point_circle" title="Nine-point circle">nine-point centers</a> of triangles BCD, ACD, ABD, ABC respectively. Then E is the <a href="/wiki/Midpoint" title="Midpoint">midpoint</a> of OH.<a href="#cite_note-Myakishev-47">[47]</a> Another remarkable line in a convex non-parallelogram quadrilateral is the <a href="/wiki/Newton_line" title="Newton line">Newton line</a>, which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the <a href="/wiki/Newton_line" title="Newton line">Newton's</a> one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.<a href="#cite_note-48">[48]</a> For any quadrilateral ABCD with points P and Q the intersections of AD and BC and AB and CD, respectively, the circles (PAB), (PCD), (QAD), and (QBC) pass through a common point M, called a Miquel point.<a href="#cite_note-49">[49]</a> For a convex quadrilateral ABCD in which E is the point of intersection of the diagonals and F is the point of intersection of the extensions of sides BC and AD, let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. Then there holds: the straight lines NK and ML intersect at point P that is located on the side AB; the straight lines NL and KM intersect at point Q that is located on the side CD. Points P and Q are called "Pascal points" formed by circle ω on sides AB and CD. <a href="#cite_note-Fraivert-50">[50]</a> <a href="#cite_note-Fraivert2-51">[51]</a> <a href="#cite_note-Fraivert3-52">[52]</a> <div class="mw-heading mw-heading2"><h2 id="Other_properties_of_convex_quadrilaterals">Other properties of convex quadrilaterals</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=24" title="Edit section: Other properties of convex quadrilaterals">edit</a>]</div> <ul><li>Let exterior squares be drawn on all sides of a quadrilateral. The segments connecting the <a href="/wiki/Centre_(geometry)#Symmetric_objects" title="Centre (geometry)">centers</a> of opposite squares are (a) equal in length, and (b) <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>. Thus these centers are the vertices of an <a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">orthodiagonal quadrilateral</a>. This is called <a href="/wiki/Van_Aubel%27s_theorem" title="Van Aubel's theorem">Van Aubel's theorem</a>.</li> <li>For any simple quadrilateral with given edge lengths, there is a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a> with the same edge lengths.<a href="#cite_note-Peter-43">[43]</a></li> <li>The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two triangles.<a href="#cite_note-53">[53]</a></li> <li>The angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> at the intersection of the diagonals satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {a^{2}+c^{2}-b^{2}-d^{2}}{2pq}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {a^{2}+c^{2}-b^{2}-d^{2}}{2pq}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ae1f5e6b5d8f8e1e68490fe9632d538b9e321f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.361ex; height:6.176ex;" alt="{\displaystyle \cos \theta ={\frac {a^{2}+c^{2}-b^{2}-d^{2}}{2pq}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953a97b9fe7d257c9666fb3cf6bf75380295e2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.362ex; height:2.009ex;" alt="{\displaystyle p,q}"> are the diagonals of the quadrilateral.<a href="#cite_note-54">[54]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Taxonomy">Taxonomy</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=25" title="Edit section: Taxonomy">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Quadrilateral_hierarchy_svg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quadrilateral_hierarchy_svg.svg/220px-Quadrilateral_hierarchy_svg.svg.png" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quadrilateral_hierarchy_svg.svg/330px-Quadrilateral_hierarchy_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quadrilateral_hierarchy_svg.svg/440px-Quadrilateral_hierarchy_svg.svg.png 2x" data-file-width="819" data-file-height="1000" /></a><figcaption>A taxonomy of quadrilaterals, using a <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a>.</figcaption></figure> A hierarchical <a href="/wiki/Taxonomy_(general)" class="mw-redirect" title="Taxonomy (general)">taxonomy</a> of quadrilaterals is illustrated by the figure to the right. Lower classes are special cases of higher classes they are connected to. Note that "trapezoid" here is referring to the North American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout. <div class="mw-heading mw-heading2"><h2 id="Skew_quadrilaterals">Skew quadrilaterals</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=26" title="Edit section: Skew quadrilaterals">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Skew_polygon" title="Skew polygon">Skew polygon</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Disphenoid_tetrahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disphenoid_tetrahedron.png/260px-Disphenoid_tetrahedron.png" decoding="async" width="260" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disphenoid_tetrahedron.png/390px-Disphenoid_tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disphenoid_tetrahedron.png/520px-Disphenoid_tetrahedron.png 2x" data-file-width="1002" data-file-height="767" /></a><figcaption>The (red) side edges of <a href="/wiki/Tetragonal_disphenoid" class="mw-redirect" title="Tetragonal disphenoid">tetragonal disphenoid</a> represent a regular zig-zag skew quadrilateral</figcaption></figure> A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as <a href="/wiki/Cyclobutane" title="Cyclobutane">cyclobutane</a> that contain a "puckered" ring of four atoms.<a href="#cite_note-55">[55]</a> Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.<a href="#cite_note-56">[56]</a> A skew quadrilateral together with its diagonals form a (possibly non-regular) <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> is removed. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=27" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Complete_quadrangle" title="Complete quadrangle">Complete quadrangle</a></li> <li><a href="/wiki/Perpendicular_bisector_construction_of_a_quadrilateral" title="Perpendicular bisector construction of a quadrilateral">Perpendicular bisector construction of a quadrilateral</a></li> <li><a href="/wiki/Saccheri_quadrilateral" title="Saccheri quadrilateral">Saccheri quadrilateral</a></li> <li><a href="/wiki/Types_of_mesh#Quadrilateral" title="Types of mesh">Types of mesh § Quadrilateral</a></li> <li><a href="/wiki/Quadrangle_(geography)" title="Quadrangle (geography)">Quadrangle (geography)</a></li> <li><a href="/wiki/Homography" title="Homography">Homography</a> - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)</li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=28" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1">^ <a href="#cite_ref-:0_1-0">a</a> <a href="#cite_ref-:0_1-1">b</a> <a href="#cite_ref-:0_1-2">c</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/quadrilaterals.html">"Quadrilaterals - 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Proceedings of the Royal Irish Academy. 4: 380–387.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Irish+Academy&rft.atitle=On+Some+Results+Obtained+by+the+Quaternion+Analysis+Respecting+the+Inscription+of+%22Gauche%22+Polygons+in+Surfaces+of+the+Second+Order&rft.volume=4&rft.pages=380-387&rft.date=1850&rft.aulast=Hamilton&rft.aufirst=William+Rowan&rft_id=http%3A%2F%2Fwww.maths.tcd.ie%2Fpub%2FHistMath%2FPeople%2FHamilton%2FGauche%2FGauche1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuadrilateral" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Quadrilateral&action=edit&section=29" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Tetragons" class="extiw" title="commons:Category:Tetragons">Tetragons</a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quadrangle,_complete">"Quadrangle, complete"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quadrangle%2C+complete&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DQuadrangle%2C_complete&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuadrilateral" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml">Quadrilaterals Formed by Perpendicular Bisectors</a>, <a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/ProjectiveQuadri.shtml">Projective Collinearity</a> and <a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/Quadrilaterals.shtml">Interactive Classification</a> of Quadrilaterals from <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/quadrilateraltoc.html">Definitions and examples of quadrilaterals</a> and <a rel="nofollow" class="external text" href="http://www.mathopenref.com/tetragon.html">Definition and properties of tetragons</a> from Mathopenref</li> <li><a rel="nofollow" class="external text" href="http://dynamicmathematicslearning.com/quad-tree-new-web.html">A (dynamic) Hierarchical Quadrilateral Tree</a> at <a rel="nofollow" class="external text" href="http://dynamicmathematicslearning.com/JavaGSPLinks.htm">Dynamic Geometry Sketches</a></li> <li><a rel="nofollow" class="external text" href="http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf">An extended classification of quadrilaterals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191230004754/http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf">Archived</a> 2019-12-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at <a rel="nofollow" class="external text" href="http://mysite.mweb.co.za/residents/profmd/homepage4.html">Dynamic Math Learning Homepage</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180825150046/http://mysite.mweb.co.za/residents/profmd/homepage4.html">Archived</a> 2018-08-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110719175018/http://mzone.mweb.co.za/residents/profmd/classify.pdf">The role and function of a hierarchical classification of quadrilaterals</a> by Michael de Villiers</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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href="/wiki/Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li> <li><a href="/wiki/Ideal_triangle" title="Ideal triangle">Ideal</a></li> <li><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a class="mw-selflink selflink">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a> </th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concave_polygon" title="Concave polygon">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Quadrilateral&oldid=1259208437">https://en.wikipedia.org/w/index.php?title=Quadrilateral&oldid=1259208437</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:4_(number)" title="Category:4 (number)">4 (number)</a></li><li><a href="/wiki/Category:Quadrilaterals" title="Category:Quadrilaterals">Quadrilaterals</a></li></ul></div><div id="mw-hidden-catlinks" 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Quadrilateral - Wikipedia