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<span>Coordinatization</span> </div> </a> <ul id="toc-Coordinatization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_versus_affine_spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Projective_versus_affine_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Projective versus affine spaces</span> </div> </a> <ul id="toc-Projective_versus_affine_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Self-duality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Self-duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Self-duality</span> </div> </a> <ul id="toc-Self-duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_of_Desargues's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_of_Desargues's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Proof of Desargues's theorem</span> </div> </a> <button aria-controls="toc-Proof_of_Desargues's_theorem-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Proof of Desargues's theorem subsection</span> </button> <ul id="toc-Proof_of_Desargues's_theorem-sublist" class="vector-toc-list"> <li id="toc-Three-dimensional_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three-dimensional_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Three-dimensional proof</span> </div> </a> <ul id="toc-Three-dimensional_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-dimensional_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-dimensional_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Two-dimensional proof</span> </div> </a> <ul id="toc-Two-dimensional_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_Pappus's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_Pappus's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Relation to Pappus's theorem</span> </div> </a> <ul id="toc-Relation_to_Pappus's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Desargues_configuration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_Desargues_configuration"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>The Desargues configuration</span> </div> </a> <ul id="toc-The_Desargues_configuration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_little_Desargues_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_little_Desargues_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>The little Desargues theorem</span> </div> </a> <ul id="toc-The_little_Desargues_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </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" 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theorem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 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-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%AF%D9%8A%D8%B2%D8%A7%D8%B1%D8%BA" title="مبرهنة ديزارغ – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة ديزارغ" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Desargues" title="Satz von Desargues – German" lang="de" hreflang="de" data-title="Satz von Desargues" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Desargues" title="Teorema de Desargues – Spanish" lang="es" hreflang="es" data-title="Teorema de Desargues" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%AF%D8%B2%D8%A7%D8%B1%DA%AF" title="قضیه دزارگ – Persian" lang="fa" hreflang="fa" data-title="قضیه دزارگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Desargues" title="Théorème de Desargues – French" lang="fr" hreflang="fr" data-title="Théorème de Desargues" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Desargues" title="Teorema de Desargues – Galician" lang="gl" hreflang="gl" data-title="Teorema de Desargues" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B0%EC%9E%90%EB%A5%B4%EA%B7%B8_%EC%A0%95%EB%A6%AC" title="데자르그 정리 – Korean" lang="ko" hreflang="ko" data-title="데자르그 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Desargues" title="Teorema di Desargues – Italian" lang="it" hreflang="it" data-title="Teorema di Desargues" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%93%D7%96%D7%A8%D7%92" title="משפט דזרג – Hebrew" lang="he" hreflang="he" data-title="משפט דזרג" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%B7%D0%B0%D1%80%D0%B3_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Дезарг теоремасы – Kazakh" lang="kk" hreflang="kk" data-title="Дезарг теоремасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B5%D0%B7%D0%B0%D1%80%D0%B3_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Дезарг теоремасы – Kyrgyz" lang="ky" hreflang="ky" data-title="Дезарг теоремасы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Desargues-t%C3%A9tel" title="Desargues-tétel – Hungarian" lang="hu" hreflang="hu" data-title="Desargues-tétel" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/T%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%94%D0%B5%D0%B7%D0%B0%D1%80%D0%B3" title="Tеорема на Дезарг – Macedonian" lang="mk" hreflang="mk" data-title="Tеорема на Дезарг" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Desargues" title="Stelling van Desargues – Dutch" lang="nl" hreflang="nl" data-title="Stelling van Desargues" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%B6%E3%83%AB%E3%82%B0%E3%81%AE%E5%AE%9A%E7%90%86" title="デザルグの定理 – Japanese" lang="ja" hreflang="ja" data-title="デザルグの定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dezarg_teoremasi" title="Dezarg teoremasi – Uzbek" lang="uz" hreflang="uz" data-title="Dezarg teoremasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Desargues%E2%80%99a" title="Twierdzenie Desargues’a – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Desargues’a" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Desargues" title="Teorema de Desargues – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Desargues" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_lui_Desargues" title="Teorema lui Desargues – Romanian" lang="ro" hreflang="ro" data-title="Teorema lui Desargues" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%94%D0%B5%D0%B7%D0%B0%D1%80%D0%B3%D0%B0" title="Теорема Дезарга – Russian" lang="ru" hreflang="ru" data-title="Теорема Дезарга" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Desarguesin_lause" title="Desarguesin lause – Finnish" lang="fi" hreflang="fi" data-title="Desarguesin lause" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Desargues_sats" title="Desargues sats – Swedish" lang="sv" hreflang="sv" data-title="Desargues sats" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Desargues_teoremi" title="Desargues teoremi – Turkish" lang="tr" hreflang="tr" data-title="Desargues teoremi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%94%D0%B5%D0%B7%D0%B0%D1%80%D0%B3%D0%B0" title="Теорема Дезарга – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема Дезарга" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Desargues" title="Định lý Desargues – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý Desargues" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AC%9B%E6%B2%99%E6%A0%BC%E5%AE%9A%E7%90%86" title="笛沙格定理 – Chinese" lang="zh" hreflang="zh" data-title="笛沙格定理" data-language-autonym="中文" data-language-local-name="Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Desargues%27_theorem&redirect=no" class="mw-redirect" title="Desargues' theorem">Desargues' theorem</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Two triangles are in perspective axially if and only if they are in perspective centrally</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Desargues_theorem_alt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Desargues_theorem_alt.svg/350px-Desargues_theorem_alt.svg.png" decoding="async" width="350" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Desargues_theorem_alt.svg/525px-Desargues_theorem_alt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Desargues_theorem_alt.svg/700px-Desargues_theorem_alt.svg.png 2x" data-file-width="578" data-file-height="430" /></a><figcaption>Perspective triangles. Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of the first condition is <a href="/wiki/Necessary_and_sufficient" class="mw-redirect" title="Necessary and sufficient">necessary and sufficient</a> for the truth of the second.</figcaption></figure> <p>In <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, <b>Desargues's theorem</b>, named after <a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desargues</a>, states: </p> <dl><dd>Two <a href="/wiki/Triangle" title="Triangle">triangles</a> are in <a href="/wiki/Perspective_(geometry)" title="Perspective (geometry)">perspective</a> <i>axially</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> they are in perspective <i>centrally</i>.</dd></dl> <p>Denote the three <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> of one triangle by <span class="texhtml"><i>a</i>, <i>b</i></span> and <span class="texhtml"><i>c</i></span>, and those of the other by <span class="texhtml"><i>A</i>, <i>B</i></span> and <span class="texhtml"><i>C</i></span>. <i>Axial <a href="/wiki/Perspectivity" title="Perspectivity">perspectivity</a></i> means that lines <span class="texhtml"><span style="text-decoration:overline;"><i>ab</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>AB</i></span></span> meet in a point, lines <span class="texhtml"><span style="text-decoration:overline;"><i>ac</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>AC</i></span></span> meet in a second point, and lines <span class="texhtml"><span style="text-decoration:overline;"><i>bc</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>BC</i></span></span> meet in a third point, and that these three points all lie on a common line called the <i>axis of perspectivity</i>. <i>Central perspectivity</i> means that the three lines <span class="texhtml"><span style="text-decoration:overline;"><i>Aa</i></span>, <span style="text-decoration:overline;"><i>Bb</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>Cc</i></span></span> are concurrent, at a point called the <i>center of perspectivity</i>. </p><p>This <a href="/wiki/Intersection_theorem" title="Intersection theorem">intersection theorem</a> is true in the usual <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following <a href="/wiki/Jean-Victor_Poncelet" title="Jean-Victor Poncelet">Jean-Victor Poncelet</a>. This results in a <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>. </p><p>Desargues's theorem is true for the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a> and for any projective space defined arithmetically from a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or <a href="/wiki/Division_ring" title="Division ring">division ring</a>; that includes any projective space of dimension greater than two or in which <a href="/wiki/Pappus%27s_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's theorem</a> holds. However, there are many "<a href="/wiki/Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a>", in which Desargues's theorem is false. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Desargues never published this theorem, but it appeared in an appendix entitled <i>Universal Method of M. Desargues for Using Perspective</i> (<i>Manière universelle de M. Desargues pour practiquer la perspective</i>) to a practical book on the use of perspective published in 1648.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> by his friend and pupil Abraham Bosse (1602–1676).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Coordinatization">Coordinatization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=2" title="Edit section: Coordinatization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring. </p> <div class="mw-heading mw-heading2"><h2 id="Projective_versus_affine_spaces">Projective versus affine spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=3" title="Edit section: Projective versus affine spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In an <a href="/wiki/Affine_space" title="Affine space">affine space</a> such as the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space. </p> <div class="mw-heading mw-heading2"><h2 id="Self-duality">Self-duality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=4" title="Edit section: Self-duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By definition, two triangles are <a href="/wiki/Perspective_(geometry)" title="Perspective (geometry)">perspective</a> if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a>. </p><p>Under the standard <a href="/wiki/Duality_(projective_geometry)" title="Duality (projective geometry)">duality of plane projective geometry</a> (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>This self-duality in the statement is due to the usual modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the <a href="/wiki/Theorem#Converse" title="Theorem">converse</a> of Desargues's theorem and was always referred to by that name.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Proof_of_Desargues's_theorem"><span id="Proof_of_Desargues.27s_theorem"></span>Proof of Desargues's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=5" title="Edit section: Proof of Desargues's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Desargues's theorem holds for projective space of any dimension over any field or division ring, and also holds for abstract projective spaces of dimension at least 3. In dimension 2 the planes for which it holds are called <a href="/wiki/Desarguesian_plane" class="mw-redirect" title="Desarguesian plane">Desarguesian planes</a> and are the same as the planes that can be given coordinates over a division ring. There are also many <a href="/wiki/Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a> where Desargues's theorem does not hold. </p> <div class="mw-heading mw-heading3"><h3 id="Three-dimensional_proof">Three-dimensional proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=6" title="Edit section: Three-dimensional proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Desargues's theorem is true for any projective space of dimension at least 3, and more generally for any projective space that can be embedded in a space of dimension at least 3. </p><p>Desargues's theorem can be stated as follows: </p> <dl><dd>If lines <span class="texhtml"><span style="text-decoration:overline;"><i>Aa</i></span>, <span style="text-decoration:overline;"><i>Bb</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>Cc</i></span></span> are concurrent (meet at a point), then</dd> <dd>the points <span class="texhtml"><span style="text-decoration:overline;"><i>AB</i></span> ∩ <span style="text-decoration:overline;"><i>ab</i></span>, <span style="text-decoration:overline;"><i>AC</i></span> ∩ <span style="text-decoration:overline;"><i>ac</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>BC</i></span> ∩ <span style="text-decoration:overline;"><i>bc</i></span></span> are <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a>.</dd></dl> <p>The points <span class="texhtml"><i>A</i>, <i>B</i>, <i>a</i></span> and <span class="texhtml"><i>b</i></span> are coplanar (lie in the same plane) because of the assumed concurrency of <span class="texhtml"><span style="text-decoration:overline;"><i>Aa</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>Bb</i></span></span>. Therefore, the lines <span class="texhtml"><span style="text-decoration:overline;"><i>AB</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>ab</i></span></span> belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point <span class="texhtml"><span style="text-decoration:overline;"><i>AB</i></span> ∩ <span style="text-decoration:overline;"><i>ab</i></span></span> belongs to both planes. By a symmetric argument, the points <span class="texhtml"><span style="text-decoration:overline;"><i>AC</i></span> ∩ <span style="text-decoration:overline;"><i>ac</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>BC</i></span> ∩ <span style="text-decoration:overline;"><i>bc</i></span></span> also exist and belong to the planes of both triangles. Since these two planes intersect in more than one point, their intersection is a line that contains all three points. </p><p>This proves Desargues's theorem if the two triangles are not contained in the same plane. If they are in the same plane, Desargues's theorem can be proved by choosing a point not in the plane, using this to lift the triangles out of the plane so that the argument above works, and then projecting back into the plane. The last step of the proof fails if the projective space has dimension less than 3, as in this case it is not possible to find a point not in the plane. </p><p><a href="/wiki/Monge%27s_theorem" title="Monge's theorem">Monge's theorem</a> also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection of two planes. </p> <div class="mw-heading mw-heading3"><h3 id="Two-dimensional_proof">Two-dimensional proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=7" title="Edit section: Two-dimensional proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As there are <a href="/wiki/Non-Desarguesian_projective_plane" class="mw-redirect" title="Non-Desarguesian projective plane">non-Desarguesian projective planes</a> in which Desargues's theorem is not true,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> some extra conditions need to be met in order to prove it. These conditions usually take the form of assuming the existence of sufficiently many <a href="/wiki/Collineation" title="Collineation">collineations</a> of a certain type, which in turn leads to showing that the underlying algebraic coordinate system must be a <a href="/wiki/Division_ring" title="Division ring">division ring</a> (skewfield).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_Pappus's_theorem"><span id="Relation_to_Pappus.27s_theorem"></span>Relation to Pappus's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=8" title="Edit section: Relation to Pappus's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Pappus%27s_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's hexagon theorem</a> states that, if a <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> <span class="texhtml"><i>AbCaBc</i></span> is drawn in such a way that vertices <span class="texhtml"><i>a</i>, <i>b</i></span> and <span class="texhtml"><i>c</i></span> lie on a line and vertices <span class="texhtml"><i>A</i>, <i>B</i></span> and <span class="texhtml"><i>C</i></span> lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called <i>Pappian</i>. <a href="#CITEREFHessenberg1905">Hessenberg (1905)</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> showed that Desargues's theorem can be deduced from three applications of Pappus's theorem.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Theorem#Converse" title="Theorem">converse</a> of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to <a href="/wiki/Wedderburn%27s_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a>, which states that all <i>finite</i> division rings are fields, all <i>finite</i> Desarguesian planes are Pappian. There is no known completely geometric proof of this fact, although <a href="#CITEREFBambergPenttila2015">Bamberg & Penttila (2015)</a> give a proof that uses only "elementary" algebraic facts (rather than the full strength of Wedderburn's little theorem). </p> <div class="mw-heading mw-heading2"><h2 id="The_Desargues_configuration">The Desargues configuration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=9" title="Edit section: The Desargues configuration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Desargues_configuration" title="Desargues configuration">Desargues configuration</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mutually-inscribed-pentagons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Mutually-inscribed-pentagons.svg/220px-Mutually-inscribed-pentagons.svg.png" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Mutually-inscribed-pentagons.svg/330px-Mutually-inscribed-pentagons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Mutually-inscribed-pentagons.svg/440px-Mutually-inscribed-pentagons.svg.png 2x" data-file-width="253" data-file-height="325" /></a><figcaption>The Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon.</figcaption></figure> <p>The ten lines involved in Desargues's theorem (six sides of triangles, the three lines <span class="texhtml"><span style="text-decoration:overline;"><i>Aa</i></span>, <span style="text-decoration:overline;"><i>Bb</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>Cc</i></span></span>, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines. Those ten points and ten lines make up the <a href="/wiki/Desargues_configuration" title="Desargues configuration">Desargues configuration</a>, an example of a <a href="/wiki/Projective_configuration" class="mw-redirect" title="Projective configuration">projective configuration</a>. Although Desargues's theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more <a href="/wiki/Symmetry" title="Symmetry">symmetric</a>: <i>any</i> of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity. </p> <div class="mw-heading mw-heading2"><h2 id="The_little_Desargues_theorem">The little Desargues theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=10" title="Edit section: The little Desargues theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well. Thus, it is the specialization of Desargues's Theorem to only the cases in which the center of perspectivity lies on the axis of perspectivity. </p><p>A <a href="/wiki/Moufang_plane" title="Moufang plane">Moufang plane</a> is a projective plane in which the little Desargues theorem is valid for every line. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Pascal%27s_theorem" title="Pascal's theorem">Pascal's theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmith1959">Smith (1959</a>, p. 307)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz1998">Katz (1998</a>, p. 461)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">(<a href="#CITEREFCoxeter1964">Coxeter 1964</a>) pp. 26–27.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">(<a href="#CITEREFCoxeter1964">Coxeter 1964</a>, pg. 19)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">The smallest examples of these can be found in <a href="#CITEREFRoomKirkpatrick1971">Room & Kirkpatrick 1971</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">(<a href="#CITEREFAlbertSandler2015">Albert & Sandler 2015</a>), (<a href="#CITEREFHughesPiper1973">Hughes & Piper 1973</a>), and (<a href="#CITEREFStevenson1972">Stevenson 1972</a>).</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">According to (<a href="#CITEREFDembowski1968">Dembowski 1968</a>, pg. 159, footnote 1), Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by <a href="#CITEREFCronheim1953">Cronheim 1953</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, p. 238, section 14.3</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 id="CITEREFAlbertSandler2015" class="citation cs2">Albert, A. Adrian; Sandler, Reuben (2015) [1968], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_E-RBQAAQBAJ"><i>An Introduction to Finite Projective Planes</i></a>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-78994-1" title="Special:BookSources/978-0-486-78994-1"><bdi>978-0-486-78994-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Finite+Projective+Planes&rft.pub=Dover&rft.date=2015&rft.isbn=978-0-486-78994-1&rft.aulast=Albert&rft.aufirst=A.+Adrian&rft.au=Sandler%2C+Reuben&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_E-RBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBambergPenttila2015" class="citation cs2">Bamberg, John; Penttila, Tim (2015), <a rel="nofollow" class="external text" href="https://research-repository.uwa.edu.au/en/publications/completing-segres-proof-of-wedderburns-little-theorem(02353184-d79b-484f-ba6f-cc32b2bab7bc).html">"Completing Segre's proof of Wedderburn's little theorem"</a>, <i><a href="/wiki/Bulletin_of_the_London_Mathematical_Society" class="mw-redirect" title="Bulletin of the London Mathematical Society">Bulletin of the London Mathematical Society</a></i>, <b>47</b> (3): 483–492, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2Fbdv021">10.1112/blms/bdv021</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123036578">123036578</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=Completing+Segre%27s+proof+of+Wedderburn%27s+little+theorem&rft.volume=47&rft.issue=3&rft.pages=483-492&rft.date=2015&rft_id=info%3Adoi%2F10.1112%2Fblms%2Fbdv021&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123036578%23id-name%3DS2CID&rft.aulast=Bamberg&rft.aufirst=John&rft.au=Penttila%2C+Tim&rft_id=https%3A%2F%2Fresearch-repository.uwa.edu.au%2Fen%2Fpublications%2Fcompleting-segres-proof-of-wedderburns-little-theorem%2802353184-d79b-484f-ba6f-cc32b2bab7bc%29.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasse2006" class="citation cs2">Casse, Rey (2006), <i>Projective Geometry: An Introduction</i>, Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-929886-6" title="Special:BookSources/0-19-929886-6"><bdi>0-19-929886-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Geometry%3A+An+Introduction&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=0-19-929886-6&rft.aulast=Casse&rft.aufirst=Rey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1964" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1964), <i>Projective Geometry</i>, Blaisdell</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Geometry&rft.pub=Blaisdell&rft.date=1964&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1969" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, Harold Scott MacDonald</a> (1969), <i>Introduction to Geometry</i> (2nd ed.), Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50458-0" title="Special:BookSources/978-0-471-50458-0"><bdi>978-0-471-50458-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0123930">0123930</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometry&rft.edition=2nd&rft.pub=Wiley&rft.date=1969&rft.isbn=978-0-471-50458-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D123930%23id-name%3DMR&rft.aulast=Coxeter&rft.aufirst=Harold+Scott+MacDonald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCronheim1953" class="citation cs2">Cronheim, Arno (1953), "A proof of Hessenberg's theorem", <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a></i>, <b>4</b> (2): 219–221, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2031794">10.2307/2031794</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2031794">2031794</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0053531">0053531</a></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+American+Mathematical+Society&rft.atitle=A+proof+of+Hessenberg%27s+theorem&rft.volume=4&rft.issue=2&rft.pages=219-221&rft.date=1953&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0053531%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2031794%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2031794&rft.aulast=Cronheim&rft.aufirst=Arno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDembowski1968" class="citation cs2">Dembowski, Peter (1968), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NJy_iTT_wGMC"><i>Finite Geometries</i></a>, Springer Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-61786-0" title="Special:BookSources/978-3-540-61786-0"><bdi>978-3-540-61786-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Geometries&rft.pub=Springer+Verlag&rft.date=1968&rft.isbn=978-3-540-61786-0&rft.aulast=Dembowski&rft.aufirst=Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNJy_iTT_wGMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHessenberg1905" class="citation cs2">Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>61</b> (2), Springer: 161–172, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01457558">10.1007/BF01457558</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1432-1807">1432-1807</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120456855">120456855</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beweis+des+Desarguesschen+Satzes+aus+dem+Pascalschen&rft.volume=61&rft.issue=2&rft.pages=161-172&rft.date=1905&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120456855%23id-name%3DS2CID&rft.issn=1432-1807&rft_id=info%3Adoi%2F10.1007%2FBF01457558&rft.aulast=Hessenberg&rft.aufirst=Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbertCohn-Vossen1952" class="citation cs2"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; <a href="/wiki/Stephan_Cohn-Vossen" class="mw-redirect" title="Stephan Cohn-Vossen">Cohn-Vossen, Stephan</a> (1952), <i>Geometry and the Imagination</i> (2nd ed.), Chelsea, pp. 119–128, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8284-1087-9" title="Special:BookSources/0-8284-1087-9"><bdi>0-8284-1087-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+and+the+Imagination&rft.pages=119-128&rft.edition=2nd&rft.pub=Chelsea&rft.date=1952&rft.isbn=0-8284-1087-9&rft.aulast=Hilbert&rft.aufirst=David&rft.au=Cohn-Vossen%2C+Stephan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughesPiper1973" class="citation cs2">Hughes, Dan; Piper, Fred (1973), <i>Projective Planes</i>, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90044-6" title="Special:BookSources/0-387-90044-6"><bdi>0-387-90044-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Planes&rft.pub=Springer-Verlag&rft.date=1973&rft.isbn=0-387-90044-6&rft.aulast=Hughes&rft.aufirst=Dan&rft.au=Piper%2C+Fred&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKárteszi1976" class="citation cs2">Kárteszi, Ferenc (1976), <i>Introduction to Finite Geometries</i>, North-Holland, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7204-2832-7" title="Special:BookSources/0-7204-2832-7"><bdi>0-7204-2832-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Finite+Geometries&rft.pub=North-Holland&rft.date=1976&rft.isbn=0-7204-2832-7&rft.aulast=K%C3%A1rteszi&rft.aufirst=Ferenc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1998" class="citation cs2">Katz, Victor J. (1998), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00katz"><i>A History of Mathematics:An Introduction</i></a></span> (2nd ed.), Reading, Mass.: Addison Wesley Longman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-321-01618-1" title="Special:BookSources/0-321-01618-1"><bdi>0-321-01618-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3AAn+Introduction&rft.place=Reading%2C+Mass.&rft.edition=2nd&rft.pub=Addison+Wesley+Longman&rft.date=1998&rft.isbn=0-321-01618-1&rft.aulast=Katz&rft.aufirst=Victor+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00katz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPambuccianSchacht2019" class="citation cs2">Pambuccian, Victor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and Desargues", in Dani, S. G.; Papadopoulos, A. (eds.), <i>Geometry in history</i>, Springer, pp. 355–399, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-13611-6" title="Special:BookSources/978-3-030-13611-6"><bdi>978-3-030-13611-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+axiomatic+destiny+of+the+theorems+of+Pappus+and+Desargues&rft.btitle=Geometry+in+history&rft.pages=355-399&rft.pub=Springer&rft.date=2019&rft.isbn=978-3-030-13611-6&rft.aulast=Pambuccian&rft.aufirst=Victor&rft.au=Schacht%2C+Celia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoomKirkpatrick1971" class="citation cs2"><a href="/wiki/Thomas_Gerald_Room" title="Thomas Gerald Room">Room, Thomas G.</a>; Kirkpatrick, P. B. (1971), <i>Miniquaternion Geometry</i>, Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-07926-8" title="Special:BookSources/0-521-07926-8"><bdi>0-521-07926-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Miniquaternion+Geometry&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1971&rft.isbn=0-521-07926-8&rft.aulast=Room&rft.aufirst=Thomas+G.&rft.au=Kirkpatrick%2C+P.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1959" class="citation cs2">Smith, David Eugene (1959), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/sourcebookinmath0000smit"><i>A Source Book in Mathematics</i></a></span>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-64690-4" title="Special:BookSources/0-486-64690-4"><bdi>0-486-64690-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematics&rft.pub=Dover&rft.date=1959&rft.isbn=0-486-64690-4&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsourcebookinmath0000smit&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStevenson1972" class="citation cs2">Stevenson, Frederick W. (1972), <i>Projective Planes</i>, W.H. Freeman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0443-9" title="Special:BookSources/0-7167-0443-9"><bdi>0-7167-0443-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Planes&rft.pub=W.H.+Freeman&rft.date=1972&rft.isbn=0-7167-0443-9&rft.aulast=Stevenson&rft.aufirst=Frederick+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoitsekhovskii2001" class="citation cs2">Voitsekhovskii, M.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Desargues_assumption">"Desargues assumption"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Desargues+assumption&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Voitsekhovskii&rft.aufirst=M.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDesargues_assumption&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADesargues%27s+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Desargues%27s_theorem&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DesarguesTheorem.html">Desargues Theorem</a> at <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml">Desargues's Theorem</a> at <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.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtml">Monge via Desargues</a> at <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="https://planetmath.org/?op=getobj&from=objects&id=4514">Proof of Desargues's theorem</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110928154549/http://math.kennesaw.edu/~mdevilli/desargues.html">Desargues's Theorem</a> at <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm">Dynamic Geometry Sketches</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Desargues%27s_theorem&oldid=1147136461">https://en.wikipedia.org/w/index.php?title=Desargues%27s_theorem&oldid=1147136461</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:Theorems_in_projective_geometry" title="Category:Theorems in projective geometry">Theorems in projective geometry</a></li><li><a href="/wiki/Category:Proof_without_words" title="Category:Proof without words">Proof without words</a></li><li><a href="/wiki/Category:Theorems_about_triangles" title="Category:Theorems about triangles">Theorems about triangles</a></li><li><a href="/wiki/Category:Euclidean_plane_geometry" title="Category:Euclidean plane geometry">Euclidean plane geometry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 March 2023, at 02:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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