Heronian triangle - Wikipedia

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parametrizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Parametrizations</span> </div> </a> <button aria-controls="toc-Parametrizations-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 Parametrizations subsection</span> </button> <ul id="toc-Parametrizations-sublist" class="vector-toc-list"> <li id="toc-Brahmagupta's_parametric_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Brahmagupta's_parametric_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Brahmagupta's parametric equation</span> </div> </a> <ul id="toc-Brahmagupta's_parametric_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler's_parametric_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler's_parametric_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Euler's parametric equation</span> </div> </a> <ul id="toc-Euler's_parametric_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Half-angle_tangent_parametrization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Half-angle_tangent_parametrization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Half-angle tangent parametrization</span> </div> </a> <ul id="toc-Half-angle_tangent_parametrization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other results</span> </div> </a> <ul id="toc-Other_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heronian_triangles_with_perfect_square_sides" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heronian_triangles_with_perfect_square_sides"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Heronian triangles with perfect square sides</span> </div> </a> <ul id="toc-Heronian_triangles_with_perfect_square_sides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equable_triangles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equable_triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Equable triangles</span> </div> </a> <ul id="toc-Equable_triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Almost-equilateral_Heronian_triangles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Almost-equilateral_Heronian_triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Almost-equilateral Heronian triangles</span> </div> </a> <ul id="toc-Almost-equilateral_Heronian_triangles-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">11</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> 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class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Heronian triangle</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. 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href="https://bs.wikipedia.org/wiki/Heronov_trougao" title="Heronov trougao – Bosnian" lang="bs" hreflang="bs" data-title="Heronov trougao" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Triangle_heroni%C3%A0" title="Triangle heronià – Catalan" lang="ca" hreflang="ca" data-title="Triangle heronià" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Heronsches_Dreieck" title="Heronsches Dreieck – German" lang="de" hreflang="de" data-title="Heronsches Dreieck" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF_%CF%84%CE%BF%CF%85_%CE%89%CF%81%CF%89%CE%BD%CE%B1" title="Τρίγωνο του Ήρωνα – Greek" lang="el" hreflang="el" data-title="Τρίγωνο του Ήρωνα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tri%C3%A1ngulo_heroniano" title="Triángulo heroniano – Spanish" lang="es" hreflang="es" data-title="Triángulo heroniano" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Triangulo_de_Herono" title="Triangulo de Herono – Esperanto" lang="eo" hreflang="eo" data-title="Triangulo de Herono" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Triangle_de_H%C3%A9ron" title="Triangle de Héron – French" lang="fr" hreflang="fr" data-title="Triangle de Héron" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Herono_trikampis" title="Herono trikampis – Lithuanian" lang="lt" hreflang="lt" data-title="Herono trikampis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Heron-driehoek" title="Heron-driehoek – Dutch" lang="nl" hreflang="nl" data-title="Heron-driehoek" 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%98%E3%83%AD%E3%83%B3%E3%81%AE%E4%B8%89%E8%A7%92%E5%BD%A2" title="ヘロンの三角形 – Japanese" lang="ja" hreflang="ja" data-title="ヘロンの三角形" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D1%80%D0%BE%D0%BD%D0%BE%D0%B2_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Геронов треугольник – Russian" lang="ru" hreflang="ru" data-title="Геронов треугольник" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Heronski_trikotnik" title="Heronski trikotnik – Slovenian" lang="sl" hreflang="sl" data-title="Heronski trikotnik" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Heronov_trougao" title="Heronov trougao – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Heronov trougao" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Heronin_kolmio" title="Heronin kolmio – Finnish" lang="fi" hreflang="fi" data-title="Heronin kolmio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%AE%E0%B8%B5%E0%B9%82%E0%B8%A3%E0%B9%80%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B8%99" title="รูปสามเหลี่ยมฮีโรเนียน – Thai" lang="th" hreflang="th" data-title="รูปสามเหลี่ยมฮีโรเนียน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA_%D0%93%D0%B5%D1%80%D0%BE%D0%BD%D0%B0" title="Трикутник Герона – Ukrainian" lang="uk" hreflang="uk" data-title="Трикутник Герона" 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/Tam_gi%C3%A1c_Heron" title="Tam giác Heron – Vietnamese" lang="vi" hreflang="vi" data-title="Tam giác Heron" 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/%E6%B5%B7%E4%BC%A6%E4%B8%89%E8%A7%92%E5%BD%A2" title="海伦三角形 – Chinese" lang="zh" hreflang="zh" data-title="海伦三角形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1613856#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Triangle whose side lengths and area are integers</div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>Heronian triangle</b> (or <b>Heron triangle</b>) is a <a href="/wiki/Triangle" title="Triangle">triangle</a> whose side lengths <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> and <a href="/wiki/Area" title="Area">area</a> <span class="texhtml mvar" style="font-style:italic;">A</span> are all positive <a href="/wiki/Integer" title="Integer">integers</a>.<sup id="cite_ref-carlson_1-0" class="reference"><a href="#cite_note-carlson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> Heronian triangles are named after <a href="/wiki/Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a>, based on their relation to <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> which Heron demonstrated with the example triangle of sides <span class="texhtml">13, 14, 15</span> and area <span class="texhtml">84</span>.<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>Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16\,A^{2}=(a+b+c)(a+b-c)(b+c-a)(c+a-b);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mspace width="thinmathspace" /> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\,A^{2}=(a+b+c)(a+b-c)(b+c-a)(c+a-b);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9fb6acbd28f9a80598a9fbbdd6b8f6c3db2021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.152ex; height:3.176ex;" alt="{\displaystyle 16\,A^{2}=(a+b+c)(a+b-c)(b+c-a)(c+a-b);}"></span></dd></dl> <p>that is, the side lengths and area of any Heronian triangle satisfy the equation, and any positive integer solution of the equation describes a Heronian triangle.<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><p>If the three side lengths are <a href="/wiki/Setwise_coprime" class="mw-redirect" title="Setwise coprime">setwise coprime</a> (meaning that the greatest common divisor of all three sides is 1), the Heronian triangle is called <i>primitive</i>. </p><p>Triangles whose side lengths and areas are all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (positive rational solutions of the above equation) are sometimes also called <i>Heronian triangles</i> or <i>rational triangles</i>;<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> in this article, these more general triangles will be called <i>rational Heronian triangles</i>. Every (integral) Heronian triangle is a rational Heronian triangle. Conversely, every rational Heronian triangle is <a href="/wiki/Similar_(geometry)" class="mw-redirect" title="Similar (geometry)">similar</a> to exactly one primitive Heronian triangle. </p><p>In any rational Heronian triangle, the three <a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">altitudes</a>, the <a href="/wiki/Circumradius" class="mw-redirect" title="Circumradius">circumradius</a>, the <a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">inradius and exradii</a>, and the <a href="/wiki/Sine_and_cosine" title="Sine and cosine">sines and cosines</a> of the three angles are also all rational numbers. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Scaling_to_primitive_triangles">Scaling to primitive triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=1" title="Edit section: Scaling to primitive triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling</a> a triangle with a factor of <span class="texhtml mvar" style="font-style:italic;">s</span> consists of multiplying its side lengths by <span class="texhtml mvar" style="font-style:italic;">s</span>; this multiplies the area by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="{\displaystyle s^{2}}"></span> and produces a <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> triangle. Scaling a rational Heronian triangle by a <a href="/wiki/Rational_number" title="Rational number">rational</a> factor produces another rational Heronian triangle. </p><p>Given a rational Heronian triangle of side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {p}{d}},{\frac {q}{d}},{\frac {r}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {p}{d}},{\frac {q}{d}},{\frac {r}{d}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54042b5f678588b1179618597f96577248288ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.802ex; height:3.676ex;" alt="{\textstyle {\frac {p}{d}},{\frac {q}{d}},{\frac {r}{d}},}"></span> the scale factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d}{\gcd(p,q,r)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d}{\gcd(p,q,r)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e762a8ec3eeb01af19c6981c2f7fdc30a4becd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.821ex; height:4.343ex;" alt="{\textstyle {\frac {d}{\gcd(p,q,r)}}}"></span> produce a rational Heronian triangle such that its side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36b3d82f06f9bbe121a2e9d45c7f97af6cb8ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\textstyle a,b,c}"></span> are setwise <a href="/wiki/Coprime_number" class="mw-redirect" title="Coprime number">coprime integers</a>. It is proved below that the area <span class="texhtml mvar" style="font-style:italic;">A</span> is an integer, and thus the triangle is a Heronian triangle. Such a triangle is often called a <i>primitive Heronian triangle.</i> </p><p>In summary, every similarity <a href="/wiki/Equivalence_class" title="Equivalence class">class</a> of rational Heronian triangles contains exactly one primitive Heronian triangle. A byproduct of the proof is that exactly one of the side lengths of a primitive Heronian triangle is an even integer. </p><p><i>Proof:</i> One has to prove that, if the side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36b3d82f06f9bbe121a2e9d45c7f97af6cb8ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\textstyle a,b,c}"></span> of a rational Heronian triangle are coprime integers, then the area <span class="texhtml mvar" style="font-style:italic;">A</span> is also an integer and exactly one of the side lengths is even. </p><p>The Diophantine equation given in the introduction shows immediately that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16A^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2329e9f7c56872161db5ed45c2e5c0a511153457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.122ex; height:2.676ex;" alt="{\displaystyle 16A^{2}}"></span> is an integer. Its square root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fe1829cfe4634233268d436c2ff501760766de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.906ex; height:2.176ex;" alt="{\displaystyle 4A}"></span> is also an integer, since the square root of an integer is either an integer or an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>. </p><p>If exactly one of the side lengths is even, all the factors in the right-hand side of the equation are even, and, by dividing the equation by <span class="texhtml">16</span>, one gets that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> are integers. </p><p>As the side lengths are supposed to be coprime, one is left with the case where one or three side lengths are odd. Supposing that <span class="texhtml mvar" style="font-style:italic;">c</span> is odd, the right-hand side of the Diophantine equation can be rewritten </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb7ac1e8db4bc5298f7b0d05239da7c7633f1b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.931ex; height:3.176ex;" alt="{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}),}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span> even. As the square of an odd integer is <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">congruent</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> modulo <span class="texhtml">4</span>, the right-hand side of the equation must be congruent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> modulo <span class="texhtml">4</span>. It is thus impossible, that one has a solution of the Diophantine equation, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16A^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2329e9f7c56872161db5ed45c2e5c0a511153457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.122ex; height:2.676ex;" alt="{\displaystyle 16A^{2}}"></span> must be the square of an integer, and the square of an integer is congruent to <span class="texhtml">0</span> or <span class="texhtml">1</span> modulo <span class="texhtml">4</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Pythagorean_triangle" class="mw-redirect" title="Pythagorean triangle">Pythagorean triangle</a> is a Heronian triangle. The side lengths of such a triangle are <a href="/wiki/Integer" title="Integer">integers</a>, by definition. In any such triangle, one of the two shorter sides has even length, so the area (the product of these two sides, divided by two) is also an integer. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Triangle-heronian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle-heronian.svg/220px-Triangle-heronian.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle-heronian.svg/330px-Triangle-heronian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle-heronian.svg/440px-Triangle-heronian.svg.png 2x" data-file-width="900" data-file-height="621" /></a><figcaption>A triangle with sidelengths <i>c</i>, <i>e</i> and <i>b</i> + <i>d</i>, and height <i>a</i>.</figcaption></figure> <p>Examples of Heronian triangles that are not right-angled are the <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangle</a> obtained by joining a Pythagorean triangle and its mirror image along a side of the right angle. Starting with the <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a> <span class="texhtml">3, 4, 5</span> this gives two Heronian triangles with side lengths <span class="texhtml">(5, 5, 6)</span> and <span class="texhtml">(5, 5, 8)</span> and area <span class="texhtml">12</span>. </p><p>More generally, given two Pythagorean triples <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,d,e)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,d,e)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b57a37dc2ee01ed30f355f8eb83d88a00af764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.406ex; height:2.843ex;" alt="{\displaystyle (a,d,e)}"></span> with largest entries <span class="texhtml mvar" style="font-style:italic;">c</span> and <span class="texhtml mvar" style="font-style:italic;">e</span>, one can join the corresponding triangles along the sides of length <span class="texhtml mvar" style="font-style:italic;">a</span> (see the figure) for getting a Heronian triangle with side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,e,b+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,e,b+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6259b35b79f2ddbe1bbd3dc877b6e9b0514d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.212ex; height:2.509ex;" alt="{\displaystyle c,e,b+d}"></span> and area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {1}{2}}a(b+d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {1}{2}}a(b+d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fccc12aff6f2aa0bd4b308b550adb9f4b2e3d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.751ex; height:3.509ex;" alt="{\textstyle {\tfrac {1}{2}}a(b+d)}"></span> (this is an integer, since the area of a Pythagorean triangle is an integer). </p><p>There are Heronian triangles that cannot be obtained by joining Pythagorean triangles. For example, the Heronian triangle of side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5,29,30}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>,</mo> <mn>29</mn> <mo>,</mo> <mn>30</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5,29,30}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068c005610e7d88444706d722c7fa8a6e1f4df3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.88ex; height:2.509ex;" alt="{\displaystyle 5,29,30}"></span> and area 72, since none of its altitudes is an integer. Such Heronian triangles are known as <em>indecomposable</em>.<sup id="cite_ref-Yiu_6-0" class="reference"><a href="#cite_note-Yiu-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> However, every Heronian triangle can be constructed from right triangles with <a href="/wiki/Rational_number" title="Rational number">rational</a> side lengths, and is thus similar to a decomposable Heronian triangle. In fact, at least one of the altitudes of a triangle is inside the triangle, and divides it into two right triangles. These triangles have rational sides, since the cosine and the sine of the angles of a Heronian triangle are rational numbers, and, with notation of the figure, one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=c\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c\sin \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96751ee7d9c4e9d46d29a4f4c904edcff051c05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.453ex; height:2.176ex;" alt="{\displaystyle a=c\sin \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=c\cos \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=c\cos \alpha ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc9ffb6d7a197dc04ecb8c1526b3cf7cdd625bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:2.509ex;" alt="{\displaystyle b=c\cos \alpha ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is the left-most angle of the triangle. </p> <div class="mw-heading mw-heading2"><h2 id="Rationality_properties">Rationality properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=3" title="Edit section: Rationality properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many quantities related to a Heronian triangle are rational numbers. In particular: </p> <ul><li>All the altitudes of a Heronian triangle are rational.<sup id="cite_ref-Somos_7-0" class="reference"><a href="#cite_note-Somos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This can be seen from the fact that the area of a triangle is half of one side times its altitude from that side, and a Heronian triangle has integer sides and area. Some Heronian triangles have three non-integer altitudes, for example the acute (15, 34, 35) with area 252 and the obtuse (5, 29, 30) with area 72. Any Heronian triangle with one or more non-integer altitudes can be scaled up by a factor equalling the least common multiple of the altitudes' denominators in order to obtain a <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> Heronian triangle with three integer altitudes.</li> <li>All the <a href="/wiki/Bisection#Perpendicular_bisectors" title="Bisection">interior perpendicular bisectors</a> of a Heronian triangle are rational: For any triangle these are given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{a}={\tfrac {2aA}{a^{2}+b^{2}-c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mi>A</mi> </mrow> <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> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{a}={\tfrac {2aA}{a^{2}+b^{2}-c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4911f1ade8464b3df4c6292c36b11004d2ae36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:14.281ex; height:4.343ex;" alt="{\displaystyle p_{a}={\tfrac {2aA}{a^{2}+b^{2}-c^{2}}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{b}={\tfrac {2bA}{a^{2}+b^{2}-c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>b</mi> <mi>A</mi> </mrow> <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> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{b}={\tfrac {2bA}{a^{2}+b^{2}-c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5fdc0b9f946cb862d80ac88c51e2599916a9d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:14.117ex; height:4.343ex;" alt="{\displaystyle p_{b}={\tfrac {2bA}{a^{2}+b^{2}-c^{2}}},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\tfrac {2cA}{a^{2}-b^{2}+c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>c</mi> <mi>A</mi> </mrow> <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> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}={\tfrac {2cA}{a^{2}-b^{2}+c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eee7c2b66d431775abee7c8633a16a0e6ae8736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:14.123ex; height:4.343ex;" alt="{\displaystyle p_{c}={\tfrac {2cA}{a^{2}-b^{2}+c^{2}}},}"></span> where the sides are <i>a</i> ≥ <i>b</i> ≥ <i>c</i> and the area is <i>A</i>;<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> in a Heronian triangle all of <i>a</i>, <i>b</i>, <i>c</i>, and <i>A</i> are integers.</li> <li>Every <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angle</a> of a Heronian triangle has a rational sine. This follows from the area formula <span class="texhtml"><i>Area</i> = (1/2)<i>ab</i> sin <i>C</i></span>, in which the area and the sides <i>a</i> and <i>b</i> are integers, and equivalently for the other interior angles.</li> <li>Every interior angle of a Heronian triangle has a rational cosine. This follows from the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a> , <span class="texhtml"><i>c</i><sup>2</sup> = <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> − 2<i>ab</i> cos <i>C</i></span>, in which the sides <i>a</i>, <i>b</i>, and <i>c</i> are integers, and equivalently for the other interior angles.</li> <li>Because all Heronian triangles have all interior angles' sines and cosines rational, this implies that the tangent, cotangent, secant, and cosecant of each interior angle is either rational or infinite.</li> <li>Half of each interior angle has a rational tangent because <span class="texhtml">tan <i>C</i>/2 = sin <i>C</i> / (1 + cos <i>C</i>)</span>, and equivalently for other interior angles. Knowledge of these half-angle tangent values is sufficient to reconstruct the side lengths of a primitive Heronian triangle (<a href="#Half-angle_tangent_parametrization">see below</a>).</li> <li>For any triangle, the angle spanned by a side as viewed from the <a href="/wiki/Circumcenter" class="mw-redirect" title="Circumcenter">center</a> of the <a href="/wiki/Circumcircle" title="Circumcircle">circumcircle</a> is twice the interior angle of the triangle vertex opposite the side. Because the half-angle tangent for each interior angle of a Heronian triangle is rational, it follows that the quarter-angle tangent of each such central angle of a Heronian triangle is rational. (Also, the quarter-angle tangents are rational for the central angles of a <a href="/wiki/Brahmagupta_quadrilateral" class="mw-redirect" title="Brahmagupta quadrilateral">Brahmagupta quadrilateral</a>, but is an unsolved problem whether this is true for all <a href="/wiki/Robbins_pentagon" title="Robbins pentagon">Robbins pentagons</a>.) <a href="/wiki/Concyclic_points#Integer_area_and_side_lengths" title="Concyclic points">The reverse is true for all cyclic polygons</a> generally; if all such central angles have rational tangents for their quarter angles then the cyclic polygon can be scaled to simultaneously have integer area, sides, and diagonals (connecting any two vertices).</li> <li>There are no Heronian triangles whose three internal angles form an arithmetic progression. This is because all plane triangles with interior angles in an arithmetic progression must have one interior angle of 60°, which does not have a rational sine.<sup id="cite_ref-Zelator_9-0" class="reference"><a href="#cite_note-Zelator-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>Any square inscribed in a Heronian triangle has rational sides: For a general triangle the <a href="/wiki/Triangle#Figures_inscribed_in_a_triangle" title="Triangle">inscribed square</a> on side of length <i>a</i> has length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2Aa}{a^{2}+2A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>A</mi> <mi>a</mi> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>A</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2Aa}{a^{2}+2A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291880a5611167e31a36d37a8344d263551aaa6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:5.87ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2Aa}{a^{2}+2A}}}"></span> where <i>A</i> is the triangle's area;<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in a Heronian triangle, both <i>A</i> and <i>a</i> are integers.</li> <li>Every Heronian triangle has a rational <a href="/wiki/Inradius" class="mw-redirect" title="Inradius">inradius</a> (radius of its inscribed circle): For a general triangle the inradius is the ratio of the area to half the perimeter, and both of these are rational in a Heronian triangle.</li> <li>Every Heronian triangle has a rational <a href="/wiki/Circumscribed_circle#Triangles" title="Circumscribed circle">circumradius</a> (the radius of its circumscribed circle): For a general triangle the circumradius equals one-fourth the product of the sides divided by the area; in a Heronian triangle the sides and area are integers.</li> <li>In a Heronian triangle the distance from the <a href="/wiki/Centroid" title="Centroid">centroid</a> to each side is rational because, for all triangles, this distance is the ratio of twice the area to three times the side length.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This can be generalized by stating that all centers associated with Heronian triangles whose <a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">barycentric coordinates</a> are rational ratios have a rational distance to each side. These centers include the <a href="/wiki/Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a>, <a href="/wiki/Orthocenter" title="Orthocenter">orthocenter</a>, <a href="/wiki/Nine-point_center" title="Nine-point center">nine-point center</a>, <a href="/wiki/Symmedian_point" class="mw-redirect" title="Symmedian point">symmedian point</a>, <a href="/wiki/Gergonne_point" class="mw-redirect" title="Gergonne point">Gergonne point</a> and <a href="/wiki/Nagel_point" title="Nagel point">Nagel point</a>.<sup id="cite_ref-ck_12-0" class="reference"><a href="#cite_note-ck-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>Every Heronian triangle can be placed on a unit-sided <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a> with each vertex at a lattice point.<sup id="cite_ref-Yiu3_13-0" class="reference"><a href="#cite_note-Yiu3-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> As a corollary, every rational Heronian triangle can be placed into a two-dimensional <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> with all rational-valued coordinates.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties_of_side_lengths">Properties of side lengths</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=4" title="Edit section: Properties of side lengths"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here are some properties of side lengths of Heronian triangles, whose side lengths are <span class="texhtml"><i>a</i>, <i>b</i>, <i>c</i></span> and area is <span class="texhtml mvar" style="font-style:italic;">A</span>. </p> <ul><li>Every primitive Heronian triangle Heronian triangle has one even and two odd sides (see <a href="#Scaling_to_primitive_triangles">§ Scaling to primitive triangles</a>). It follows that a Heronian triangle has either one or three sides of even length,<sup id="cite_ref-Buchholz1_14-0" class="reference"><a href="#cite_note-Buchholz1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.3">: p.3 </span></sup> and that the perimeter of a primitive Heronian triangle is always an even number.<sup id="cite_ref-Friche_15-0" class="reference"><a href="#cite_note-Friche-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li>There are no equilateral Heronian triangles, since a primitive Heronian triangle has one even side length and two odd side lengths.<sup id="cite_ref-Somos_7-1" class="reference"><a href="#cite_note-Somos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>The area of a Heronian triangle is always divisible by 6.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Friche_15-1" class="reference"><a href="#cite_note-Friche-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li>There are no Heronian triangles with a side length of either 1 or 2.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-carlson_1-1" class="reference"><a href="#cite_note-carlson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>There exist an infinite number of primitive Heronian triangles with one side length equal to a given <span class="texhtml"><i>a</i></span>, provided that <span class="texhtml"><i>a </i>> 2</span>.<sup id="cite_ref-carlson_1-2" class="reference"><a href="#cite_note-carlson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>The semiperimeter <span class="texhtml"><i>s</i></span> of a Heronian triangle cannot be prime (as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(s-a)(s-b)(s-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(s-a)(s-b)(s-c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6043dcf911ae525f57c458ba60e87a9ede7e8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.545ex; height:2.843ex;" alt="{\displaystyle s(s-a)(s-b)(s-c)}"></span> is the square of the area, and the area is an integer, if <span class="texhtml mvar" style="font-style:italic;">s</span> were prime, it would divide another factor; this is impossible as these factors are all less than <span class="texhtml"><i>s</i></span>).</li> <li>In a Heronian triangles that has no integer altitude (<a href="#Properties_of_side_lengths">indecomposable</a> and non-Pythagorean), all side lengths have a prime factor of the form <span class="texhtml">4<i>k</i>+1</span>.<sup id="cite_ref-Yiu_6-1" class="reference"><a href="#cite_note-Yiu-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In a primitive Pythagoran triangle, all <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> of the hypotenuse have the form <span class="texhtml">4<i>k</i>+1</span>. A decomposable Heronian triangle must have two sides that are the hypotenuse of a Pythagorean triangle, and thus two sides that have prime factors of the form <span class="texhtml">4<i>k</i>+1</span>. There may also be prime factors of the form <span class="texhtml">4<i>k</i>+3</span>, since the Pythagorean components of a decomposable Heronian triangle need not to be primitive, even if the Heronian triangle is primitive. In summary, all Heronian triangles have at least one side that is divisible by a prime of the form <span class="texhtml">4<i>k</i>+1</span>.</li> <li>There are no Heronian triangles whose side lengths form a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a>.<sup id="cite_ref-Buchholz_18-0" class="reference"><a href="#cite_note-Buchholz-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li>If any two sides (but not three) of a Heronian triangle have a common factor, that factor must be the sum of two squares.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Parametrizations">Parametrizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=5" title="Edit section: Parametrizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equation</a> or <i>parametrization</i> of Heronian triangles consists of an expression of the side lengths and area of a triangle as functions—typically <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> – of some parameters, such that the triangle is Heronian if and only if the parameters satisfy some constraints—typically, to be positive integers satisfying some inequalities. It is also generally required that all Heronian triangles can be obtained up to a scaling for some values of the parameters, and that these values are unique, if an order on the sides of the triangle is specified. </p><p>The first such parametrization was discovered by <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (598-668 A.D.), who did not prove that all Heronian triangles can be generated by the parametrization. In the 18th century, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> provided another parametrization and proved that it generates all Heronian triangles. These parametrizations are described in the next two subsections. </p><p>In the third subsection, a rational parametrization—that is a parametrization where the parameters are positive <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>—is naturally derived from properties of Heronian triangles. Both Brahmagupta's and Euler's parametrizations can be recovered from this rational parametrization by <a href="/wiki/Clearing_denominators" title="Clearing denominators">clearing denominators</a>. This provides a proof that Brahmagupta's and Euler's parametrizations generate all Heronian triangles. </p> <div class="mw-heading mw-heading3"><h3 id="Brahmagupta's_parametric_equation"><span id="Brahmagupta.27s_parametric_equation"></span>Brahmagupta's parametric equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=6" title="Edit section: Brahmagupta's parametric equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Indian mathematician <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (598-668 A.D.) discovered the following <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equations</a> for generating Heronian triangles,<sup id="cite_ref-Kurz_20-0" class="reference"><a href="#cite_note-Kurz-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> but did not prove that every <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a> class of Heronian triangles can be obtained this way.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2023)">citation needed</span></a></i>]</sup> </p><p>For three positive integers <span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">k</span> that are <a href="/wiki/Coprime_integers#Coprimality_in_sets" title="Coprime integers">setwise coprime</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(m,n,k)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(m,n,k)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a463d8a28114608403adedd9f2b21a9c8791e36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.272ex; height:2.843ex;" alt="{\displaystyle \gcd(m,n,k)=1}"></span>) and satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mn>k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> <mo>></mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn>k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e861bcd3dab33ce6cc254e2e596bbd333580ee1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.799ex; height:2.676ex;" alt="{\displaystyle mn>k^{2}}"></span> (to guarantee positive side lengths) and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b02f25e62da7fe3162ac80446437cdc1c0fd341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}"></span></span> (for uniqueness): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&=n(m^{2}+k^{2}),&s-a&={\tfrac {1}{2}}(b+c-a)=n(mn-k^{2}),\\b&=m(n^{2}+k^{2}),&s-b&={\tfrac {1}{2}}(c+a-b)=m(mn-k^{2}),\\c&=(m+n)(mn-k^{2}),&s-c&={\tfrac {1}{2}}(a+b-c)=(m+n)k^{2},\\&&s&={\tfrac {1}{2}}(a+b+c)=mn(m+n),\\A&=mnk(m+n)(mn-k^{2}),&r&=k(mn-k^{2}),\\\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>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <mi></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>b</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>b</mi> </mtd> <mtd> <mi></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>c</mi> <mo>+</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>c</mi> </mtd> <mtd> <mi></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>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mi>s</mi> </mtd> <mtd> <mi></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>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>n</mi> <mi>k</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=n(m^{2}+k^{2}),&s-a&={\tfrac {1}{2}}(b+c-a)=n(mn-k^{2}),\\b&=m(n^{2}+k^{2}),&s-b&={\tfrac {1}{2}}(c+a-b)=m(mn-k^{2}),\\c&=(m+n)(mn-k^{2}),&s-c&={\tfrac {1}{2}}(a+b-c)=(m+n)k^{2},\\&&s&={\tfrac {1}{2}}(a+b+c)=mn(m+n),\\A&=mnk(m+n)(mn-k^{2}),&r&=k(mn-k^{2}),\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2d51e001502cf46fc1b51ecd03b0d4fe4f8765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:70.744ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}a&=n(m^{2}+k^{2}),&s-a&={\tfrac {1}{2}}(b+c-a)=n(mn-k^{2}),\\b&=m(n^{2}+k^{2}),&s-b&={\tfrac {1}{2}}(c+a-b)=m(mn-k^{2}),\\c&=(m+n)(mn-k^{2}),&s-c&={\tfrac {1}{2}}(a+b-c)=(m+n)k^{2},\\&&s&={\tfrac {1}{2}}(a+b+c)=mn(m+n),\\A&=mnk(m+n)(mn-k^{2}),&r&=k(mn-k^{2}),\\\end{aligned}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">s</span> is the semiperimeter, <span class="texhtml mvar" style="font-style:italic;">A</span> is the area, and <span class="texhtml mvar" style="font-style:italic;">r</span> is the inradius. </p><p>The resulting Heronian triangle is not always primitive, and a scaling may be needed for getting the corresponding primitive triangle. For example, taking <span class="texhtml"><i>m</i> = 36</span>, <span class="texhtml"><i>n</i> = 4</span> and <span class="texhtml"><i>k</i> = 3</span> produces a triangle with <span class="texhtml"><i>a</i> = 5220</span>, <span class="texhtml"><i>b</i> = 900</span> and <span class="texhtml"><i>c</i> = 5400</span>, which is similar to the <span class="texhtml">(5, 29, 30)</span> Heronian triangle with a proportionality factor of <span class="texhtml">180</span>. </p><p>The fact that the generated triangle is not primitive is an obstacle for using this parametrization for generating all Heronian triangles with size lengths less than a given bound (since the size of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f6ce94f77257d63d831132c290291f174e1733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.599ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c)}"></span> cannot be predicted.<sup id="cite_ref-Kurz_20-1" class="reference"><a href="#cite_note-Kurz-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Euler's_parametric_equation"><span id="Euler.27s_parametric_equation"></span>Euler's parametric equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=7" title="Edit section: Euler's parametric equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following method of generating all Heronian triangles was discovered by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>,<sup id="cite_ref-FOOTNOTEDickson1920193_21-0" class="reference"><a href="#cite_note-FOOTNOTEDickson1920193-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> who was the first to provably parametrize all such triangles. </p><p>For four positive integers <span class="texhtml mvar" style="font-style:italic;">m</span> coprime to <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">p</span> coprime to <span class="texhtml mvar" style="font-style:italic;">q</span> <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd {(m,n)}=\gcd {(p,q)}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd {(m,n)}=\gcd {(p,q)}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfc9591fa126ead42859ef9cae8a5d9d2f2f52e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.469ex; height:2.843ex;" alt="{\displaystyle \gcd {(m,n)}=\gcd {(p,q)}=1}"></span>)</span> satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mp>nq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>p</mi> <mo>></mo> <mi>n</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mp>nq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7a9d24fb80d8e587dd0d76df65793099aec962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.773ex; height:2.176ex;" alt="{\displaystyle mp>nq}"></span> (to guarantee positive side lengths): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&=mn(p^{2}+q^{2}),&s-a&=mq(mp-nq),\\b&=pq(m^{2}+n^{2}),&s-b&=np(mp-nq),\\c&=(mq+np)(mp-nq),&s-c&=nq(mq+np),\\&&s&=mp(mq+np),\\A&=mnpq(mq+np)(mp-nq),&r&=nq(mp-nq),\\\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>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>n</mi> <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> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>p</mi> <mo>−</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mi>q</mi> <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> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>p</mi> <mo>−</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>q</mi> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>p</mi> <mo>−</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>q</mi> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>q</mi> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>q</mi> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>p</mi> <mo>−</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>p</mi> <mo>−</mo> <mi>n</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=mn(p^{2}+q^{2}),&s-a&=mq(mp-nq),\\b&=pq(m^{2}+n^{2}),&s-b&=np(mp-nq),\\c&=(mq+np)(mp-nq),&s-c&=nq(mq+np),\\&&s&=mp(mq+np),\\A&=mnpq(mq+np)(mp-nq),&r&=nq(mp-nq),\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de23e4cb5c73053ae2f7f9938e96afbf11d3ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:59.646ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}a&=mn(p^{2}+q^{2}),&s-a&=mq(mp-nq),\\b&=pq(m^{2}+n^{2}),&s-b&=np(mp-nq),\\c&=(mq+np)(mp-nq),&s-c&=nq(mq+np),\\&&s&=mp(mq+np),\\A&=mnpq(mq+np)(mp-nq),&r&=nq(mp-nq),\\\end{aligned}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">s</span> is the semiperimeter, <span class="texhtml mvar" style="font-style:italic;">A</span> is the area, and <span class="texhtml mvar" style="font-style:italic;">r</span> is the inradius. </p><p>Even when <span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">p</span>, and <span class="texhtml mvar" style="font-style:italic;">q</span> are pairwise coprime, the resulting Heronian triangle may not be primitive. In particular, if <span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">p</span>, and <span class="texhtml mvar" style="font-style:italic;">q</span> are all odd, the three side lengths are even. It is also possible that <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> have a common divisor other than <span class="texhtml">2</span>. For example, with <span class="texhtml"><i>m</i> = 2</span>, <span class="texhtml"><i>n</i> = 1</span>, <span class="texhtml"><i>p</i> = 7</span>, and <span class="texhtml"><i>q</i> = 4</span>, one gets <span class="texhtml">(<i>a</i>, <i>b</i>, <i>c</i>) = (130, 140, 150)</span>, where each side length is a multiple of <span class="texhtml">10</span>; the corresponding primitive triple is <span class="texhtml">(13, 14, 15)</span>, which can also be obtained by dividing the triple resulting from <span class="texhtml"><i>m</i> = 2, <i>n</i> = 1, <i>p</i> = 3, <i>q</i> = 2</span> by two, then exchanging <span class="texhtml"><i>b</i></span> and <span class="texhtml"><i>c</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Half-angle_tangent_parametrization">Half-angle tangent parametrization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=8" title="Edit section: Half-angle tangent parametrization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Triangle-tikz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle-tikz.svg/220px-Triangle-tikz.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle-tikz.svg/330px-Triangle-tikz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle-tikz.svg/440px-Triangle-tikz.svg.png 2x" data-file-width="163" data-file-height="157" /></a><figcaption>A triangle with side lengths and interior angles labeled as in the text</figcaption></figure> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4e864488dd5c0492166b3424066681acf38617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.563ex; height:2.509ex;" alt="{\displaystyle a,b,c>0}"></span> be the side lengths of a triangle, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }"></span> be the interior angles opposite these sides, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t=\tan {\frac {\alpha }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t=\tan {\frac {\alpha }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91457027b8925d37a6081c2920f52270f3e28bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.22ex; height:3.176ex;" alt="{\textstyle t=\tan {\frac {\alpha }{2}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u=\tan {\frac {\beta }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u=\tan {\frac {\beta }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbd3f346d3bb5a7eb25dc6dca9b3793587c820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.6ex; height:4.009ex;" alt="{\textstyle u=\tan {\frac {\beta }{2}},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle v=\tan {\frac {\gamma }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v=\tan {\frac {\gamma }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25c22ab89364326178b4b4b1321d3724a5e8646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.702ex; height:3.509ex;" alt="{\textstyle v=\tan {\frac {\gamma }{2}}}"></span> be the half-angle tangents. The values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,u,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,u,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fe2d669e3c1595cf04d203a0f5c029bf074852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.365ex; height:2.343ex;" alt="{\displaystyle t,u,v}"></span> are all positive and satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tu+uv+vt=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mi>v</mi> <mo>+</mo> <mi>v</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tu+uv+vt=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6645680a7ca726d7aa75ba216bb03d6a65cb8f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.536ex; height:2.343ex;" alt="{\displaystyle tu+uv+vt=1}"></span>; this "triple tangent identity" is the half-angle tangent version of the fundamental triangle identity written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\alpha }{2}}+{\frac {\beta }{2}}+{\frac {\gamma }{2}}={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\alpha }{2}}+{\frac {\beta }{2}}+{\frac {\gamma }{2}}={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7eddbacc865ab89f9e972f8140ed5ec8e97bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.952ex; height:4.009ex;" alt="{\textstyle {\frac {\alpha }{2}}+{\frac {\beta }{2}}+{\frac {\gamma }{2}}={\frac {\pi }{2}}}"></span> radians (that is, 90°), as can be proved using the <a href="/wiki/Angle_addition_formulas" class="mw-redirect" title="Angle addition formulas">addition formula for tangents</a>. By the <a href="/wiki/Law_of_sines" title="Law of sines">laws of sines</a> and <a href="/wiki/Law_of_cosines" title="Law of cosines">cosines</a>, all of the sines and the cosines of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }"></span> are rational numbers if the triangle is a rational Heronian triangle and, because a <a href="/wiki/Tangent_half-angle_formula" title="Tangent half-angle formula">half-angle tangent is a rational function of the sine and cosine</a>, it follows that the half-angle tangents are also rational. </p><p>Conversely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,u,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,u,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fe2d669e3c1595cf04d203a0f5c029bf074852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.365ex; height:2.343ex;" alt="{\displaystyle t,u,v}"></span> are positive rational numbers such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tu+uv+vt=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mi>v</mi> <mo>+</mo> <mi>v</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tu+uv+vt=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e351abd82acd95d08882ea51982102545d7d16f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.182ex; height:2.509ex;" alt="{\displaystyle tu+uv+vt=1,}"></span> it can be seen that they are the half-angle tangents of the interior angles of a class of similar Heronian triangles.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tu+uv+vt=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mi>v</mi> <mo>+</mo> <mi>v</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tu+uv+vt=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6645680a7ca726d7aa75ba216bb03d6a65cb8f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.536ex; height:2.343ex;" alt="{\displaystyle tu+uv+vt=1}"></span> can be rearranged to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle v={\frac {1-tu}{t+u}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v={\frac {1-tu}{t+u}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675e1ccb99eab0e65a13444171abd49484808ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.344ex; height:3.843ex;" alt="{\textstyle v={\frac {1-tu}{t+u}},}"></span> and the restriction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v>0}"></span> requires <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tu<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>u</mi> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tu<1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47a4095cedeb9006b5185a3fbc98bcb2d24fabc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.077ex; height:2.176ex;" alt="{\displaystyle tu<1.}"></span> Thus there is a bijection between the similarity classes of rational Heronian triangles and the pairs of positive rational numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4acceddd581eb2b8b2b75abd7465fca74fb6e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle (t,u)}"></span> whose product is less than <span class="texhtml">1</span>. </p><p>To make this bijection explicit, one can choose, as a specific member of the similarity class, the triangle inscribed in a unit-diameter circle with side lengths equal to the sines of the opposite angles:<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&=\sin \alpha ={\frac {2t}{1+t^{2}}},&s-a={\frac {2u(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]b&=\sin \beta ={\frac {2u}{1+u^{2}}},&s-b={\frac {2t(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]c&=\sin \gamma ={\frac {2(t+u)(1-tu)}{(1+t^{2})(1+u^{2})}},&s-c={\frac {2tu(t+u)}{(1+t^{2})(1+u^{2})}},\\[5mu]&&s={\frac {2(t+u)}{(1+t^{2})(1+u^{2})}},\\A&={\frac {4tu(t+u)(1-tu)}{(1+t^{2})^{2}(1+u^{2})^{2}}},&r={\frac {2tu(1-tu)}{(1+t^{2})(1+u^{2})}},\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="0.578em 0.578em 0.578em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>t</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</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> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</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> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=\sin \alpha ={\frac {2t}{1+t^{2}}},&s-a={\frac {2u(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]b&=\sin \beta ={\frac {2u}{1+u^{2}}},&s-b={\frac {2t(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]c&=\sin \gamma ={\frac {2(t+u)(1-tu)}{(1+t^{2})(1+u^{2})}},&s-c={\frac {2tu(t+u)}{(1+t^{2})(1+u^{2})}},\\[5mu]&&s={\frac {2(t+u)}{(1+t^{2})(1+u^{2})}},\\A&={\frac {4tu(t+u)(1-tu)}{(1+t^{2})^{2}(1+u^{2})^{2}}},&r={\frac {2tu(1-tu)}{(1+t^{2})(1+u^{2})}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de7e22c0ee7991d810beae4ff9a454d8f2ef1f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.838ex; width:60.932ex; height:34.843ex;" alt="{\displaystyle {\begin{aligned}a&=\sin \alpha ={\frac {2t}{1+t^{2}}},&s-a={\frac {2u(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]b&=\sin \beta ={\frac {2u}{1+u^{2}}},&s-b={\frac {2t(1-tu)}{(1+t^{2})(1+u^{2})}},\\[5mu]c&=\sin \gamma ={\frac {2(t+u)(1-tu)}{(1+t^{2})(1+u^{2})}},&s-c={\frac {2tu(t+u)}{(1+t^{2})(1+u^{2})}},\\[5mu]&&s={\frac {2(t+u)}{(1+t^{2})(1+u^{2})}},\\A&={\frac {4tu(t+u)(1-tu)}{(1+t^{2})^{2}(1+u^{2})^{2}}},&r={\frac {2tu(1-tu)}{(1+t^{2})(1+u^{2})}},\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2c4193212526a50585182f301e85e2f1cdfde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.571ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"></span> is the semiperimeter, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\tfrac {1}{2}}ab\sin \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</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> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\tfrac {1}{2}}ab\sin \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4f75fb39f105e72900932f0ecc0dba42e8a539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.619ex; height:3.509ex;" alt="{\displaystyle A={\tfrac {1}{2}}ab\sin \gamma }"></span> is the area, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {\tfrac {(s-a)(s-b)(s-c)}{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\sqrt {\tfrac {(s-a)(s-b)(s-c)}{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9f8ca83ae7907af20a7cee25177a665738c1b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:19.581ex; height:4.843ex;" alt="{\displaystyle r={\sqrt {\tfrac {(s-a)(s-b)(s-c)}{s}}}}"></span> is the inradius, and all these values are rational because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> are rational. </p><p>To obtain an (integral) Heronian triangle, the denominators of <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> <a href="/wiki/Clearing_denominators" title="Clearing denominators">must be cleared</a>. There are several ways to do this. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=m/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=m/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c6647c898570d95e9a35b910e7a3cfa8453078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.536ex; height:2.843ex;" alt="{\displaystyle t=m/n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=p/q,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=p/q,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219fbafce55c4169ea11b9523500acf3252a88dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.476ex; height:2.843ex;" alt="{\displaystyle u=p/q,}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(m,n)=\gcd(p,q)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(m,n)=\gcd(p,q)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b4f83de863d0f698420ec70e723e4ef746878e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.695ex; height:2.843ex;" alt="{\displaystyle \gcd(m,n)=\gcd(p,q)=1}"></span> (<a href="/wiki/Irreducible_fraction" title="Irreducible fraction">irreducible fractions</a>), and the triangle is scaled up by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(m^{2}+n^{2})(p^{2}+q^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>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 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 {\tfrac {1}{2}}(m^{2}+n^{2})(p^{2}+q^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0d1d9a66d58c787a53aeaa277321289912d588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.505ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(m^{2}+n^{2})(p^{2}+q^{2}),}"></span> the result is Euler's parametrization. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=m/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=m/k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7411dd591e438548d3ada12ae91b9ef2abeb826c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.352ex; height:2.843ex;" alt="{\displaystyle t=m/k}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=n/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=n/k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316afbb5186709d71d215d2bde742add67ba446e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.197ex; height:2.843ex;" alt="{\displaystyle u=n/k}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(m,n,k)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(m,n,k)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a463d8a28114608403adedd9f2b21a9c8791e36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.272ex; height:2.843ex;" alt="{\displaystyle \gcd(m,n,k)=1}"></span> (lowest common denomimator), and the triangle is scaled up by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>k</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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560f42a2617bd2174a79de3c6e7409244dca3a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:3.176ex;" alt="{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k,}"></span> the result is similar but not quite identical to Brahmagupta's parametrization. If, instead, this is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac33aabb19fba3d5d9b2e6008f61658ca2a3af3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.165ex; height:2.843ex;" alt="{\displaystyle 1/t}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17afc6bd3fb0ce6d610204dee2a869a3ff007466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/u}"></span> that are reduced to the lowest common denominator, that is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=k/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=k/m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9692759fcedc42c2f401e040e2f5c47d3d81a516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.352ex; height:2.843ex;" alt="{\displaystyle t=k/m}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=k/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=k/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7e1f9f850f4172e6c6d78aa66efd5ce376dd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.197ex; height:2.843ex;" alt="{\displaystyle u=k/n}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(m,n,k)=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(m,n,k)=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489662fb66859e0fc9e1806d7cca6fd135515629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.919ex; height:2.843ex;" alt="{\displaystyle \gcd(m,n,k)=1,}"></span> then one gets exactly Brahmagupta's parametrization by scaling up the triangle by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>k</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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bb5314f9eae3ca5efc48389f238520fcafc543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:3.176ex;" alt="{\displaystyle (k^{2}+m^{2})(k^{2}+n^{2})/2k.}"></span> </p><p>This proves that either parametrization generates all Heronian triangles. </p> <div class="mw-heading mw-heading2"><h2 id="Other_results">Other results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=9" title="Edit section: Other results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFKurz2008">Kurz (2008)</a> has derived fast algorithms for generating Heronian triangles. </p><p>There are infinitely many primitive and indecomposable non-Pythagorean Heronian triangles with integer values for the <a href="/wiki/Inradius" class="mw-redirect" title="Inradius">inradius</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and all three of the <a href="/wiki/Inradius" class="mw-redirect" title="Inradius">exradii</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{a},r_{b},r_{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{a},r_{b},r_{c})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d7038bf3aec63be7e05df818ac4ac6be22c30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle (r_{a},r_{b},r_{c})}"></span>, including the ones generated by<sup id="cite_ref-Yiu1_24-0" class="reference"><a href="#cite_note-Yiu1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm. 4">: Thm. 4 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&=5(5n^{2}+n-1),&r_{a}&=5n+3,\\b&=(5n+3)(5n^{2}-4n+1),&r_{b}&=5n^{2}+n-1,\\c&=(5n-2)(5n^{2}+6n+2),&r_{c}&=(5n-2)(5n+3)(5n^{2}+n-1),\\&&r&=5n-2,\\A&=(5n-2)(5n+3)(5n^{2}+n-1)=r_{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>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=5(5n^{2}+n-1),&r_{a}&=5n+3,\\b&=(5n+3)(5n^{2}-4n+1),&r_{b}&=5n^{2}+n-1,\\c&=(5n-2)(5n^{2}+6n+2),&r_{c}&=(5n-2)(5n+3)(5n^{2}+n-1),\\&&r&=5n-2,\\A&=(5n-2)(5n+3)(5n^{2}+n-1)=r_{c}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/741a5007e97384f461282635246f320a26404b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.261ex; width:82.667ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}a&=5(5n^{2}+n-1),&r_{a}&=5n+3,\\b&=(5n+3)(5n^{2}-4n+1),&r_{b}&=5n^{2}+n-1,\\c&=(5n-2)(5n^{2}+6n+2),&r_{c}&=(5n-2)(5n+3)(5n^{2}+n-1),\\&&r&=5n-2,\\A&=(5n-2)(5n+3)(5n^{2}+n-1)=r_{c}.\end{aligned}}}"></span></dd></dl> <p>There are infinitely many Heronian triangles that can be placed on a lattice such that not only are the vertices at lattice points, as holds for all Heronian triangles, but additionally the centers of the incircle and excircles are at lattice points.<sup id="cite_ref-Yiu1_24-1" class="reference"><a href="#cite_note-Yiu1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm. 5">: Thm. 5 </span></sup> </p><p>See also <a href="/wiki/Integer_triangle#Heronian_triangles" title="Integer triangle">Integer triangle § Heronian triangles</a> for parametrizations of some types of Heronian triangles. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_2">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=10" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The list of primitive integer Heronian triangles, sorted by area and, if this is the same, by <a href="/wiki/Perimeter" title="Perimeter">perimeter</a>, starts as in the following table. "Primitive" means that the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the three side lengths equals 1. </p> <table class="wikitable" style="text-align:right;"> <tbody><tr> <th>Area </th> <th>Perimeter </th> <th>side length b+d </th> <th>side length e </th> <th>side length c </th></tr> <tr> <td>6 </td> <td>12 </td> <td>5 </td> <td>4 </td> <td>3 </td></tr> <tr> <td>12 </td> <td>16 </td> <td>6 </td> <td>5 </td> <td>5 </td></tr> <tr> <td>12 </td> <td>18 </td> <td>8 </td> <td>5 </td> <td>5 </td></tr> <tr> <td>24 </td> <td>32 </td> <td>15 </td> <td>13 </td> <td>4 </td></tr> <tr> <td>30 </td> <td>30 </td> <td>13 </td> <td>12 </td> <td>5 </td></tr> <tr> <td>36 </td> <td>36 </td> <td>17 </td> <td>10 </td> <td>9 </td></tr> <tr> <td>36 </td> <td>54 </td> <td>26 </td> <td>25 </td> <td>3 </td></tr> <tr> <td>42 </td> <td>42 </td> <td>20 </td> <td>15 </td> <td>7 </td></tr> <tr> <td>60 </td> <td>36 </td> <td>13 </td> <td>13 </td> <td>10 </td></tr> <tr> <td>60 </td> <td>40 </td> <td>17 </td> <td>15 </td> <td>8 </td></tr> <tr> <td>60 </td> <td>50 </td> <td>24 </td> <td>13 </td> <td>13 </td></tr> <tr> <td>60 </td> <td>60 </td> <td>29 </td> <td>25 </td> <td>6 </td></tr> <tr> <td>66 </td> <td>44 </td> <td>20 </td> <td>13 </td> <td>11 </td></tr> <tr> <td>72 </td> <td>64 </td> <td>30 </td> <td>29 </td> <td>5 </td></tr> <tr> <td>84 </td> <td>42 </td> <td>15 </td> <td>14 </td> <td>13 </td></tr> <tr> <td>84 </td> <td>48 </td> <td>21 </td> <td>17 </td> <td>10 </td></tr> <tr> <td>84 </td> <td>56 </td> <td>25 </td> <td>24 </td> <td>7 </td></tr> <tr> <td>84 </td> <td>72 </td> <td>35 </td> <td>29 </td> <td>8 </td></tr> <tr> <td>90 </td> <td>54 </td> <td>25 </td> <td>17 </td> <td>12 </td></tr> <tr> <td>90 </td> <td>108 </td> <td>53 </td> <td>51 </td> <td>4 </td></tr> <tr> <td>114 </td> <td>76 </td> <td>37 </td> <td>20 </td> <td>19 </td></tr> <tr> <td>120 </td> <td>50 </td> <td>17 </td> <td>17 </td> <td>16 </td></tr> <tr> <td>120 </td> <td>64 </td> <td>30 </td> <td>17 </td> <td>17 </td></tr> <tr> <td>120 </td> <td>80 </td> <td>39 </td> <td>25 </td> <td>16 </td></tr> <tr> <td>126 </td> <td>54 </td> <td>21 </td> <td>20 </td> <td>13 </td></tr> <tr> <td>126 </td> <td>84 </td> <td>41 </td> <td>28 </td> <td>15 </td></tr> <tr> <td>126 </td> <td>108 </td> <td>52 </td> <td>51 </td> <td>5 </td></tr> <tr> <td>132 </td> <td>66 </td> <td>30 </td> <td>25 </td> <td>11 </td></tr> <tr> <td>156 </td> <td>78 </td> <td>37 </td> <td>26 </td> <td>15 </td></tr> <tr> <td>156 </td> <td>104 </td> <td>51 </td> <td>40 </td> <td>13 </td></tr> <tr> <td>168 </td> <td>64 </td> <td>25 </td> <td>25 </td> <td>14 </td></tr> <tr> <td>168 </td> <td>84 </td> <td>39 </td> <td>35 </td> <td>10 </td></tr> <tr> <td>168 </td> <td>98 </td> <td>48 </td> <td>25 </td> <td>25 </td></tr> <tr> <td>180 </td> <td>80 </td> <td>37 </td> <td>30 </td> <td>13 </td></tr> <tr> <td>180 </td> <td>90 </td> <td>41 </td> <td>40 </td> <td>9 </td></tr> <tr> <td>198 </td> <td>132 </td> <td>65 </td> <td>55 </td> <td>12 </td></tr> <tr> <td>204 </td> <td>68 </td> <td>26 </td> <td>25 </td> <td>17 </td></tr> <tr> <td>210 </td> <td>70 </td> <td>29 </td> <td>21 </td> <td>20 </td></tr> <tr> <td>210 </td> <td>70 </td> <td>28 </td> <td>25 </td> <td>17 </td></tr> <tr> <td>210 </td> <td>84 </td> <td>39 </td> <td>28 </td> <td>17 </td></tr> <tr> <td>210 </td> <td>84 </td> <td>37 </td> <td>35 </td> <td>12 </td></tr> <tr> <td>210 </td> <td>140 </td> <td>68 </td> <td>65 </td> <td>7 </td></tr> <tr> <td>210 </td> <td>300 </td> <td>149 </td> <td>148 </td> <td>3 </td></tr> <tr> <td>216 </td> <td>162 </td> <td>80 </td> <td>73 </td> <td>9 </td></tr> <tr> <td>234 </td> <td>108 </td> <td>52 </td> <td>41 </td> <td>15 </td></tr> <tr> <td>240 </td> <td>90 </td> <td>40 </td> <td>37 </td> <td>13 </td></tr> <tr> <td>252 </td> <td>84 </td> <td>35 </td> <td>34 </td> <td>15 </td></tr> <tr> <td>252 </td> <td>98 </td> <td>45 </td> <td>40 </td> <td>13 </td></tr> <tr> <td>252 </td> <td>144 </td> <td>70 </td> <td>65 </td> <td>9 </td></tr> <tr> <td>264 </td> <td>96 </td> <td>44 </td> <td>37 </td> <td>15 </td></tr> <tr> <td>264 </td> <td>132 </td> <td>65 </td> <td>34 </td> <td>33 </td></tr> <tr> <td>270 </td> <td>108 </td> <td>52 </td> <td>29 </td> <td>27 </td></tr> <tr> <td>288 </td> <td>162 </td> <td>80 </td> <td>65 </td> <td>17 </td></tr> <tr> <td>300 </td> <td>150 </td> <td>74 </td> <td>51 </td> <td>25 </td></tr> <tr> <td>300 </td> <td>250 </td> <td>123 </td> <td>122 </td> <td>5 </td></tr> <tr> <td>306 </td> <td>108 </td> <td>51 </td> <td>37 </td> <td>20 </td></tr> <tr> <td>330 </td> <td>100 </td> <td>44 </td> <td>39 </td> <td>17 </td></tr> <tr> <td>330 </td> <td>110 </td> <td>52 </td> <td>33 </td> <td>25 </td></tr> <tr> <td>330 </td> <td>132 </td> <td>61 </td> <td>60 </td> <td>11 </td></tr> <tr> <td>330 </td> <td>220 </td> <td>109 </td> <td>100 </td> <td>11 </td></tr> <tr> <td>336 </td> <td>98 </td> <td>41 </td> <td>40 </td> <td>17 </td></tr> <tr> <td>336 </td> <td>112 </td> <td>53 </td> <td>35 </td> <td>24 </td></tr> <tr> <td>336 </td> <td>128 </td> <td>61 </td> <td>52 </td> <td>15 </td></tr> <tr> <td>336 </td> <td>392 </td> <td>195 </td> <td>193 </td> <td>4 </td></tr> <tr> <td>360 </td> <td>90 </td> <td>36 </td> <td>29 </td> <td>25 </td></tr> <tr> <td>360 </td> <td>100 </td> <td>41 </td> <td>41 </td> <td>18 </td></tr> <tr> <td>360 </td> <td>162 </td> <td>80 </td> <td>41 </td> <td>41 </td></tr> <tr> <td>390 </td> <td>156 </td> <td>75 </td> <td>68 </td> <td>13 </td></tr> <tr> <td>396 </td> <td>176 </td> <td>87 </td> <td>55 </td> <td>34 </td></tr> <tr> <td>396 </td> <td>198 </td> <td>97 </td> <td>90 </td> <td>11 </td></tr> <tr> <td>396 </td> <td>242 </td> <td>120 </td> <td>109 </td> <td>13 </td></tr></tbody></table> <p>The list of primitive Heronian triangles whose sides do not exceed 6,000,000 has been computed by <a href="#CITEREFKurz2008">Kurz (2008)</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Heronian_triangles_with_perfect_square_sides">Heronian triangles with perfect square sides</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=11" title="Edit section: Heronian triangles with perfect square sides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Heronian triangles with perfect square sides are related to the <a href="/wiki/Perfect_cuboid" class="mw-redirect" title="Perfect cuboid">Perfect cuboid</a> problem. As of February 2021, only two <i>primitive</i> Heronian triangles with perfect square sides are known: </p><p>(1853², 4380², 4427², Area=32918611718880), published in 2013.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>(11789², 68104² , 68595², Area=284239560530875680), published in 2018.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Equable_triangles">Equable triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=12" title="Edit section: Equable triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A shape is called <a href="/wiki/Equable_shape" title="Equable shape">equable</a> if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17),<sup id="cite_ref-FOOTNOTEDickson1920199_27-0" class="reference"><a href="#cite_note-FOOTNOTEDickson1920199-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> though only four of them are primitive. </p> <div class="mw-heading mw-heading2"><h2 id="Almost-equilateral_Heronian_triangles">Almost-equilateral Heronian triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=13" title="Edit section: Almost-equilateral Heronian triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the area of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> with rational sides is an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, no equilateral triangle is Heronian. However, a sequence of isosceles Heronian triangles that are "almost equilateral" can be developed from the duplication of <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">right-angled triangles</a>, in which the hypotenuse is almost twice as long as one of the legs. The first few examples of these almost-equilateral triangles are listed in the following table (sequence <span class="nowrap external"><a href="//oeis.org/A102341" class="extiw" title="oeis:A102341">A102341</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <table class="wikitable" style="table-layout: fixed; width: 500px;"> <tbody><tr> <th colspan="3">Side length</th> <th rowspan="2">Area </th></tr> <tr> <th><i>a</i></th> <th><i>b</i>=<i>a</i></th> <th><i>c</i> </th></tr> <tr align="right"> <td>5</td> <td>5</td> <td>6</td> <td>12 </td></tr> <tr align="right"> <td>17</td> <td>17</td> <td>16</td> <td>120 </td></tr> <tr align="right"> <td>65</td> <td>65</td> <td>66</td> <td>1848 </td></tr> <tr align="right"> <td>241</td> <td>241</td> <td>240</td> <td>25080 </td></tr> <tr align="right"> <td>901</td> <td>901</td> <td>902</td> <td>351780 </td></tr> <tr align="right"> <td>3361</td> <td>3361</td> <td>3360</td> <td>4890480 </td></tr> <tr align="right"> <td>12545</td> <td>12545</td> <td>12546</td> <td>68149872 </td></tr> <tr align="right"> <td>46817</td> <td>46817</td> <td>46816</td> <td>949077360 </td></tr></tbody></table> <p>There is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form <i>n</i> − 1, <i>n</i>, <i>n</i> + 1. A method for generating all solutions to this problem based on <a href="/wiki/Continued_fraction" title="Continued fraction">continued fractions</a> was described in 1864 by <a href="/wiki/Edward_Sang" title="Edward Sang">Edward Sang</a>,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and in 1880 <a href="/wiki/Reinhold_Hoppe" title="Reinhold Hoppe">Reinhold Hoppe</a> gave a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a> for the solutions.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> The first few examples of these almost-equilateral triangles are listed in the following table (sequence <span class="nowrap external"><a href="//oeis.org/A003500" class="extiw" title="oeis:A003500">A003500</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <table class="wikitable" style="table-layout: fixed; width: 500px;"> <tbody><tr> <th colspan="3">Side length</th> <th rowspan="2">Area</th> <th rowspan="2">Inradius </th></tr> <tr> <th><i>n</i> − 1</th> <th><i>n</i></th> <th><i>n</i> + 1 </th></tr> <tr align="right"> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>1 </td></tr> <tr align="right"> <td>13</td> <td>14</td> <td>15</td> <td>84</td> <td>4 </td></tr> <tr align="right"> <td>51</td> <td>52</td> <td>53</td> <td>1170</td> <td>15 </td></tr> <tr align="right"> <td>193</td> <td>194</td> <td>195</td> <td>16296</td> <td>56 </td></tr> <tr align="right"> <td>723</td> <td>724</td> <td>725</td> <td>226974</td> <td>209 </td></tr> <tr align="right"> <td>2701</td> <td>2702</td> <td>2703</td> <td>3161340</td> <td>780 </td></tr> <tr align="right"> <td>10083</td> <td>10084</td> <td>10085</td> <td>44031786</td> <td>2911 </td></tr> <tr align="right"> <td>37633</td> <td>37634</td> <td>37635</td> <td>613283664</td> <td>10864 </td></tr></tbody></table> <p>Subsequent values of <i>n</i> can be found by multiplying the previous value by 4, then subtracting the value prior to that one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc.), thus: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{t}=4n_{t-1}-n_{t-2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{t}=4n_{t-1}-n_{t-2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4afa40cab401265b7bddae01af8266d65a475d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.998ex; height:2.509ex;" alt="{\displaystyle n_{t}=4n_{t-1}-n_{t-2}\,,}"></span></dd></dl> <p>where <i>t</i> denotes any row in the table. This is a <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas sequence</a>. Alternatively, the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2+{\sqrt {3}})^{t}+(2-{\sqrt {3}})^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+{\sqrt {3}})^{t}+(2-{\sqrt {3}})^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661c90615ee9af84b1abef9c50ce6ddb9404a0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.313ex; height:3.009ex;" alt="{\displaystyle (2+{\sqrt {3}})^{t}+(2-{\sqrt {3}})^{t}}"></span> generates all <i>n</i> for positive integers <i>t</i>. Equivalently, let <i>A</i> = area and <i>y</i> = inradius, then, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}(n-1)^{2}+n^{2}+(n+1)^{2}{\big )}^{2}-2{\big (}(n-1)^{4}+n^{4}+(n+1)^{4}{\big )}=(6ny)^{2}=(4A)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>n</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}(n-1)^{2}+n^{2}+(n+1)^{2}{\big )}^{2}-2{\big (}(n-1)^{4}+n^{4}+(n+1)^{4}{\big )}=(6ny)^{2}=(4A)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6860e7e0b13ccf896e87e5db778fe290b459f7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:77.163ex; height:3.676ex;" alt="{\displaystyle {\big (}(n-1)^{2}+n^{2}+(n+1)^{2}{\big )}^{2}-2{\big (}(n-1)^{4}+n^{4}+(n+1)^{4}{\big )}=(6ny)^{2}=(4A)^{2}}"></span></dd></dl> <p>where {<i>n</i>, <i>y</i>} are solutions to <i>n</i><sup>2</sup> − 12<i>y</i><sup>2</sup> = 4. A small transformation <i>n</i> = <i>2x</i> yields a conventional <a href="/wiki/Pell_equation" class="mw-redirect" title="Pell equation">Pell equation</a> <i>x</i><sup>2</sup> − 3<i>y</i><sup>2</sup> = 1, the solutions of which can then be derived from the <a href="/wiki/Continued_fraction" title="Continued fraction">regular continued fraction</a> expansion for <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>The variable <i>n</i> is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\sqrt {2+2k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\sqrt {2+2k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3409c41a6a819d0d21cebac56406b2bfededa7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.806ex; height:3.009ex;" alt="{\displaystyle n={\sqrt {2+2k}}}"></span>, where <i>k</i> is 7, 97, 1351, 18817, .... The numbers in this sequence have the property that <i>k</i> consecutive integers have integral <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </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=Heronian_triangle&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Heronian_tetrahedron" title="Heronian tetrahedron">Heronian tetrahedron</a></li> <li><a href="/wiki/Brahmagupta_quadrilateral" class="mw-redirect" title="Brahmagupta quadrilateral">Brahmagupta quadrilateral</a></li> <li><a href="/wiki/Brahmagupta_triangle" title="Brahmagupta triangle">Brahmagupta triangle</a></li> <li><a href="/wiki/Robbins_pentagon" title="Robbins pentagon">Robbins pentagon</a></li> <li><a href="/wiki/Integer_triangle#Heronian_triangles" title="Integer triangle">Integer triangle#Heronian triangles</a></li></ul> <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=Heronian_triangle&action=edit&section=15" title="Edit section: References"><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 mw-references-columns"><ol class="references"> <li id="cite_note-carlson-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-carlson_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-carlson_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-carlson_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><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 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(January 1998), <a rel="nofollow" class="external text" href="http://www.maa.org/mathdl/CMJ/methodoflastresort.pdf">"The Brahmagupta Triangles"</a> <span class="cs1-format">(PDF)</span>, <i>College Mathematics Journal</i>, <b>29</b> (1): 13–17, <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%2F2687630">10.2307/2687630</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/2687630">2687630</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=College+Mathematics+Journal&rft.atitle=The+Brahmagupta+Triangles&rft.volume=29&rft.issue=1&rft.pages=13-17&rft.date=1998-01&rft_id=info%3Adoi%2F10.2307%2F2687630&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2687630%23id-name%3DJSTOR&rft.aulast=Beauregard&rft.aufirst=Raymond+A.&rft.au=Suryanarayan%2C+E.+R.&rft_id=http%3A%2F%2Fwww.maa.org%2Fmathdl%2FCMJ%2Fmethodoflastresort.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSastry2001" class="citation journal cs1">Sastry, K. R. S. (2001). <a rel="nofollow" class="external text" href="https://forumgeom.fau.edu/FG2001volume1/FG200104.pdf">"Heron triangles: A Gergonne-Cevian-and-median perspective"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i>. <b>1</b> (2001): 17–24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+Geometricorum&rft.atitle=Heron+triangles%3A+A+Gergonne-Cevian-and-median+perspective&rft.volume=1&rft.issue=2001&rft.pages=17-24&rft.date=2001&rft.aulast=Sastry&rft.aufirst=K.+R.+S.&rft_id=https%3A%2F%2Fforumgeom.fau.edu%2FFG2001volume1%2FFG200104.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></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">The sides and area of any triangle satisfy the Diophantine equation obtained by squaring both sides of Heron's formula; see <a href="/wiki/Heron%27s_formula#Proofs" title="Heron's formula">Heron's formula § Proofs</a>. Conversely, consider a solution of the equation where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c,A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c,A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5651ce2b2ad81bb7cc36ea17fc656f29d474989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.888ex; height:2.843ex;" alt="{\displaystyle (a,b,c,A)}"></span> are all positive integers. It corresponds to a valid triangle <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> is satisfied, that is, if the three integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b-c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b-c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8974ba3d9436e58b126a6b46aed6196e43d3046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.562ex; height:2.509ex;" alt="{\displaystyle a+b-c,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+c-a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+c-a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b657e8b729077efa853363adb4bc02f5247824e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.562ex; height:2.509ex;" alt="{\displaystyle b+c-a,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c+a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c+a-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29eec77df854901126fc0cc731d91b03f05c2f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.915ex; height:2.343ex;" alt="{\displaystyle c+a-b}"></span> are all positive. This is necessarily true in this case: if any of these sums were negative or zero, the other two would be positive and the right-hand side of the equation would thus be negative or zero and could not possibly equal the left-hand side <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16\,A^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mspace width="thinmathspace" /> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\,A^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47688f74984173092bdbfc807241e8910e1ab149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.156ex; height:3.009ex;" alt="{\displaystyle 16\,A^{2},}"></span> which is positive.</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"><span class="citation mathworld" id="Reference-Mathworld-Heronian_Triangle"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HeronianTriangle.html">"Heronian Triangle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Heronian+Triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHeronianTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span></span> </li> <li id="cite_note-Yiu-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Yiu_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Yiu_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYiu2008" class="citation cs2">Yiu, Paul (2008), <a rel="nofollow" class="external text" href="http://math.fau.edu/yiu/Southern080216.pdf"><i>Heron triangles which cannot be decomposed into two integer right triangles</i></a> <span class="cs1-format">(PDF)</span>, 41st Meeting of Florida Section of Mathematical Association of America</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Heron+triangles+which+cannot+be+decomposed+into+two+integer+right+triangles&rft.pub=41st+Meeting+of+Florida+Section+of+Mathematical+Association+of+America&rft.date=2008&rft.aulast=Yiu&rft.aufirst=Paul&rft_id=http%3A%2F%2Fmath.fau.edu%2Fyiu%2FSouthern080216.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Somos-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Somos_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Somos_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSomos2014" class="citation web cs1"><a href="/wiki/Michael_Somos" title="Michael Somos">Somos, M.</a> (December 2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211220133826/http://grail.eecs.csuohio.edu/~somos/rattri.html">"Rational triangles"</a>. Archived from <a rel="nofollow" class="external text" href="http://grail.eecs.csuohio.edu/~somos/rattri.html">the original</a> on 2021-12-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-11-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Rational+triangles&rft.date=2014-12&rft.aulast=Somos&rft.aufirst=M.&rft_id=http%3A%2F%2Fgrail.eecs.csuohio.edu%2F~somos%2Frattri.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></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">Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i> 13, 53−59: Theorem 2.</span> </li> <li id="cite_note-Zelator-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zelator_9-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://arxiv.org/ftp/arxiv/papers/0803/0803.3778.pdf">Zelator, K., "Triangle Angles and Sides in Progression and the diophantine equation x<sup>2</sup>+3y<sup>2</sup>=z<sup>2</sup>", <i>Cornell Univ. archive</i>, 2008</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i> 71(4), 1998, 278–284.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="/wiki/Clark_Kimberling" title="Clark Kimberling">Clark Kimberling</a>, "Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers", <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i>, 10 (2010), 135−139. <a rel="nofollow" class="external free" href="http://forumgeom.fau.edu/FG2010volume10/FG201015index.html">http://forumgeom.fau.edu/FG2010volume10/FG201015index.html</a></span> </li> <li id="cite_note-ck-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-ck_12-0">^</a></b></span> <span class="reference-text">Clark Kimberling's <a href="/wiki/Encyclopedia_of_Triangle_Centers" title="Encyclopedia of Triangle Centers">Encyclopedia of Triangle Centers</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://faculty.evansville.edu/ck6/encyclopedia/ETC.html">"Encyclopedia of Triangle Centers"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2012-06-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Encyclopedia+of+Triangle+Centers&rft_id=https%3A%2F%2Ffaculty.evansville.edu%2Fck6%2Fencyclopedia%2FETC.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Yiu3-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yiu3_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYiu2001" class="citation journal cs1">Yiu, Paul (2001). "Heronian triangles are lattice triangles". <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>108</b> (3): 261–263. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2001.11919751">10.1080/00029890.2001.11919751</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Heronian+triangles+are+lattice+triangles&rft.volume=108&rft.issue=3&rft.pages=261-263&rft.date=2001&rft_id=info%3Adoi%2F10.1080%2F00029890.2001.11919751&rft.aulast=Yiu&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Buchholz1-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Buchholz1_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchholzMacDougall" class="citation journal cs1">Buchholz, Ralph H.; MacDougall, James A. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2007.05.005">"Cyclic Polygons with Rational Sides and Area"</a>. <i><a href="/wiki/Journal_of_Number_Theory" title="Journal of Number Theory">Journal of Number Theory</a></i>. <b>128</b> (1): 17–48. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2007.05.005">10.1016/j.jnt.2007.05.005</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Cyclic+Polygons+with+Rational+Sides+and+Area&rft.volume=128&rft.issue=1&rft.pages=17-48&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2007.05.005&rft.aulast=Buchholz&rft.aufirst=Ralph+H.&rft.au=MacDougall%2C+James+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.jnt.2007.05.005&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Friche-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-Friche_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Friche_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFricke2002" class="citation arxiv cs1">Fricke, Jan (2002-12-21). "On Heron Simplices and Integer Embedding". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0112239">math/0112239</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+Heron+Simplices+and+Integer+Embedding&rft.date=2002-12-21&rft_id=info%3Aarxiv%2Fmath%2F0112239&rft.aulast=Fricke&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><i>Proof</i>. One can suppose that the Heronian triangle is primitive. The right-hand side of the Diophantine equation can be rewritten as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/405d51e3e6793971af506da0b93b2114571db0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.931ex; height:3.176ex;" alt="{\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}).}"></span> If an odd length is chosen for <span class="texhtml mvar" style="font-style:italic;">c</span>, all squares are odd, and therefore of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8k+1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8k+1;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48eda9ddec98e7eaae00484eca588a8fead3ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.023ex; height:2.509ex;" alt="{\displaystyle 8k+1;}"></span> and the two differences are multiple of <span class="texhtml">8<span class="texhtml"></span></span>. So <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16A^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2329e9f7c56872161db5ed45c2e5c0a511153457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.122ex; height:2.676ex;" alt="{\displaystyle 16A^{2}}"></span> is multiple of <span class="texhtml">64</span>, and <span class="texhtml mvar" style="font-style:italic;">A</span> is even. For the divisibility by three, one chooses <span class="texhtml mvar" style="font-style:italic;">c</span> as non-multiple of <span class="texhtml">3</span> (the triangle is supposed to be primitive). If one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span> is not a multiple of <span class="texhtml mvar" style="font-style:italic;">3</span>, the corresponding factor is a nultiple of <span class="texhtml mvar" style="font-style:italic;">3</span> (since the square of a non-multiple of <span class="texhtml mvar" style="font-style:italic;">3</span> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3k+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6128d1598e2bdb1907ec6b40269511683d1ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.377ex; height:2.343ex;" alt="{\displaystyle 3k+1}"></span>), and this implies that <span class="texhtml mvar" style="font-style:italic;">3</span> is a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16A^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc7bf9f399c64db448b4ec011c15725c89c1dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.769ex; height:2.676ex;" alt="{\displaystyle 16A^{2}.}"></span> Otherwise, <span class="texhtml mvar" style="font-style:italic;">3</span> would divide both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07930023935ac089aba48ae39e3d7e13026a5eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.715ex; height:2.509ex;" alt="{\displaystyle a-b,}"></span> and the right-hand side of the Diophantine would not be the square of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8609cc54a6dbec7c804057e44c489d3d9aa8aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.509ex;" alt="{\displaystyle 4A,}"></span> as being congruent to minus times a square modulo <span class="texhtml">3</span>. So this last case is impossible.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><i>Proof</i>. Supposing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq b\geq c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mi>b</mi> <mo>≥</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq b\geq c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc67aa1fe8a7909904710527de731c3512d0dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.078ex; height:2.509ex;" alt="{\displaystyle a\geq b\geq c,}"></span> the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq a\leq b+c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq a\leq b+c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b63801fb7f8f0b2ae60dd3654eccdccc8e34f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.916ex; height:2.343ex;" alt="{\displaystyle b\leq a\leq b+c.}"></span> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c2402da4b692aa69d97e804956eeb37114ca2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.915ex; height:2.509ex;" alt="{\displaystyle c=1,}"></span> this implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a87d9bccbbd750d94a977aa90d98d60210d0c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a=b,}"></span> and the condition that there is exactly one even side length cannot be fulfilled. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579f2227f31e2d3ad9140f3e247065b0fe1eec60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.915ex; height:2.509ex;" alt="{\displaystyle c=2,}"></span> one has two even side lengths if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b48fae1bb036a37ce1efb9056b673ec534fb17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.976ex; height:2.343ex;" alt="{\displaystyle a=b+1.}"></span> So, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a87d9bccbbd750d94a977aa90d98d60210d0c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a=b,}"></span> and the Diophantine equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16A^{2}=16(a^{2}-1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A^{2}=16(a^{2}-1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcf1bb6053320ef41f259131af330d986445386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.289ex; height:3.176ex;" alt="{\displaystyle 16A^{2}=16(a^{2}-1),}"></span> which is impossible for two positive integers.</span> </li> <li id="cite_note-Buchholz-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Buchholz_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchholzMacDougall1999" class="citation journal cs1">Buchholz, Ralph H.; MacDougall, James A. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0004972700032883">"Heron Quadrilaterals with sides in Arithmetic or Geometric progression"</a>. <i>Bulletin of the Australian Mathematical Society</i>. <b>59</b> (2): 263–269. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0004972700032883">10.1017/s0004972700032883</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1959.13%2F803798">1959.13/803798</a></span>.</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+Australian+Mathematical+Society&rft.atitle=Heron+Quadrilaterals+with+sides+in+Arithmetic+or+Geometric+progression&rft.volume=59&rft.issue=2&rft.pages=263-269&rft.date=1999&rft_id=info%3Ahdl%2F1959.13%2F803798&rft_id=info%3Adoi%2F10.1017%2Fs0004972700032883&rft.aulast=Buchholz&rft.aufirst=Ralph+H.&rft.au=MacDougall%2C+James+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252Fs0004972700032883&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlichfeldt1896–1897" class="citation journal cs1">Blichfeldt, H. F. (1896–1897). "On Triangles with Rational Sides and Having Rational Areas". <i>Annals of Mathematics</i>. <b>11</b> (1/6): 57–60. <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%2F1967214">10.2307/1967214</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/1967214">1967214</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+Triangles+with+Rational+Sides+and+Having+Rational+Areas&rft.volume=11&rft.issue=1%2F6&rft.pages=57-60&rft.date=1896%2F1897&rft_id=info%3Adoi%2F10.2307%2F1967214&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1967214%23id-name%3DJSTOR&rft.aulast=Blichfeldt&rft.aufirst=H.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Kurz-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kurz_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kurz_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurz2008" class="citation journal cs1">Kurz, Sascha (2008). <a rel="nofollow" class="external text" href="https://eudml.org/doc/11461">"On the generation of Heronian triangles"</a>. <i>Serdica Journal of Computing</i>. <b>2</b> (2): 181–196. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1401.6150">1401.6150</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014arXiv1401.6150K">2014arXiv1401.6150K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.55630%2Fsjc.2008.2.181-196">10.55630/sjc.2008.2.181-196</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=2473583">2473583</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:16060132">16060132</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Serdica+Journal+of+Computing&rft.atitle=On+the+generation+of+Heronian+triangles&rft.volume=2&rft.issue=2&rft.pages=181-196&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16060132%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2014arXiv1401.6150K&rft_id=info%3Aarxiv%2F1401.6150&rft_id=info%3Adoi%2F10.55630%2Fsjc.2008.2.181-196&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2473583%23id-name%3DMR&rft.aulast=Kurz&rft.aufirst=Sascha&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F11461&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span>.</span> </li> <li id="cite_note-FOOTNOTEDickson1920193-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDickson1920193_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDickson1920">Dickson 1920</a>, p. 193.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheney1929" class="citation journal cs1">Cheney, William Fitch Jr. (1929). <a rel="nofollow" class="external text" href="http://math.fau.edu/yiu/PSRM2015/yiu/New%20Folder%20(4)/Heron%20Triangles/Cheney.pdf">"Heronian Triangles"</a> <span class="cs1-format">(PDF)</span>. <i>American Mathematical Monthly</i>. <b>36</b> (1): 22–28. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1929.11986902">10.1080/00029890.1929.11986902</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Heronian+Triangles&rft.volume=36&rft.issue=1&rft.pages=22-28&rft.date=1929&rft_id=info%3Adoi%2F10.1080%2F00029890.1929.11986902&rft.aulast=Cheney&rft.aufirst=William+Fitch+Jr.&rft_id=http%3A%2F%2Fmath.fau.edu%2Fyiu%2FPSRM2015%2Fyiu%2FNew%2520Folder%2520%284%29%2FHeron%2520Triangles%2FCheney.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKocikSolecki2009" class="citation journal cs1">Kocik, Jerzy; Solecki, Andrzej (2009). <a rel="nofollow" class="external text" href="http://lagrange.math.siu.edu/Kocik/triangle/monthlyTriangle.pdf">"Disentangling a triangle"</a> <span class="cs1-format">(PDF)</span>. <i>American Mathematical Monthly</i>. <b>116</b> (3): 228–237. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2009.11920932">10.1080/00029890.2009.11920932</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:28155804">28155804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Disentangling+a+triangle&rft.volume=116&rft.issue=3&rft.pages=228-237&rft.date=2009&rft_id=info%3Adoi%2F10.1080%2F00029890.2009.11920932&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28155804%23id-name%3DS2CID&rft.aulast=Kocik&rft.aufirst=Jerzy&rft.au=Solecki%2C+Andrzej&rft_id=http%3A%2F%2Flagrange.math.siu.edu%2FKocik%2Ftriangle%2FmonthlyTriangle.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-Yiu1-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Yiu1_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Yiu1_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Zhou, Li, "Primitive Heronian Triangles With Integer Inradius and Exradii", <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i> 18, 2018, 71-77. <a rel="nofollow" class="external free" href="http://forumgeom.fau.edu/FG2018volume18/FG201811.pdf">http://forumgeom.fau.edu/FG2018volume18/FG201811.pdf</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStănicăSarkarSen_GuptaMaitra2013" class="citation journal cs1">Stănică, Pantelimon; Sarkar, Santanu; Sen Gupta, Sourav; Maitra, Subhamoy; Kar, Nirupam (2013). "Counting Heron triangles with constraints". <i>Integers</i>. <b>13</b>: Paper No. A3, 17pp. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10945%2F38838">10945/38838</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=3083465">3083465</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=Counting+Heron+triangles+with+constraints&rft.volume=13&rft.pages=Paper+No.+A3%2C+17pp&rft.date=2013&rft_id=info%3Ahdl%2F10945%2F38838&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3083465%23id-name%3DMR&rft.aulast=St%C4%83nic%C4%83&rft.aufirst=Pantelimon&rft.au=Sarkar%2C+Santanu&rft.au=Sen+Gupta%2C+Sourav&rft.au=Maitra%2C+Subhamoy&rft.au=Kar%2C+Nirupam&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;e7c96423.1804&S=b">"LISTSERV - NMBRTHRY Archives - LISTSERV.NODAK.EDU"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=LISTSERV+-+NMBRTHRY+Archives+-+LISTSERV.NODAK.EDU&rft_id=https%3A%2F%2Flistserv.nodak.edu%2Fcgi-bin%2Fwa.exe%3FA2%3DNMBRTHRY%3Be7c96423.1804%26S%3Db&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEDickson1920199-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDickson1920199_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDickson1920">Dickson 1920</a>, p. 199.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarkowitz1981" class="citation cs2">Markowitz, L. (1981), "Area = Perimeter", <i>The Mathematics Teacher</i>, <b>74</b> (3): 222–3, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.74.3.0222">10.5951/MT.74.3.0222</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=Area+%3D+Perimeter&rft.volume=74&rft.issue=3&rft.pages=222-3&rft.date=1981&rft_id=info%3Adoi%2F10.5951%2FMT.74.3.0222&rft.aulast=Markowitz&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSang1864" class="citation cs2"><a href="/wiki/Edward_Sang" title="Edward Sang">Sang, Edward</a> (1864), "On the theory of commensurables", <i>Transactions of the Royal Society of Edinburgh</i>, <b>23</b> (3): 721–760, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0080456800020019">10.1017/s0080456800020019</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:123752318">123752318</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Royal+Society+of+Edinburgh&rft.atitle=On+the+theory+of+commensurables&rft.volume=23&rft.issue=3&rft.pages=721-760&rft.date=1864&rft_id=info%3Adoi%2F10.1017%2Fs0080456800020019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123752318%23id-name%3DS2CID&rft.aulast=Sang&rft.aufirst=Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uXAxAQAAMAAJ&pg=PA734">p. 734</a>.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGould1973" class="citation cs2">Gould, H. W. (February 1973), <a rel="nofollow" class="external text" href="http://www.mathstat.dal.ca/FQ/Scanned/11-1/gould.pdf">"A triangle with integral sides and area"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>, <b>11</b> (1): 27–39</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fibonacci+Quarterly&rft.atitle=A+triangle+with+integral+sides+and+area&rft.volume=11&rft.issue=1&rft.pages=27-39&rft.date=1973-02&rft.aulast=Gould&rft.aufirst=H.+W.&rft_id=http%3A%2F%2Fwww.mathstat.dal.ca%2FFQ%2FScanned%2F11-1%2Fgould.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span>.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichardson2007" class="citation cs2">Richardson, William H. (2007), <a rel="nofollow" class="external text" href="http://www.math.wichita.edu/~richardson/heronian/heronian.html"><i>Super-Heronian Triangles</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Super-Heronian+Triangles&rft.date=2007&rft.aulast=Richardson&rft.aufirst=William+H.&rft_id=http%3A%2F%2Fwww.math.wichita.edu%2F~richardson%2Fheronian%2Fheronian.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Online Encyclopedia of Integer Sequences, <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A011943" class="extiw" title="oeis:A011943">A011943</a></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heronian_triangle&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarmichael1915" class="citation book cs1"><a href="/wiki/Robert_Daniel_Carmichael" title="Robert Daniel Carmichael">Carmichael, Robert Daniel</a> (1915). <a rel="nofollow" class="external text" href="https://archive.org/details/diophantineanaly00carmuoft/page/1">"I. Introduction. Rational Triangles. Method of Infinite Descent"</a>. <i>Diophantine Analysis</i>. Wiley. pp. 1–23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=I.+Introduction.+Rational+Triangles.+Method+of+Infinite+Descent&rft.btitle=Diophantine+Analysis&rft.pages=1-23&rft.pub=Wiley&rft.date=1915&rft.aulast=Carmichael&rft.aufirst=Robert+Daniel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiophantineanaly00carmuoft%2Fpage%2F1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickson1920" class="citation book cs1"><a href="/wiki/Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson, Leonard Eugene</a> (1920). <a rel="nofollow" class="external text" href="https://archive.org/details/historyoftheoryo02dick_0/page/191">"V. Triangles, Quadrilaterals, and Tetrahedra with Rational Sides"</a>. <i><a href="/wiki/History_of_the_Theory_of_Numbers" title="History of the Theory of Numbers">History of the Theory of Numbers</a>, Volume II: Diophantine Analysis</i>. Carnegie Institution of Washington. pp. 191–224.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=V.+Triangles%2C+Quadrilaterals%2C+and+Tetrahedra+with+Rational+Sides&rft.btitle=History+of+the+Theory+of+Numbers%2C+Volume+II%3A+Diophantine+Analysis&rft.pages=191-224&rft.pub=Carnegie+Institution+of+Washington&rft.date=1920&rft.aulast=Dickson&rft.aufirst=Leonard+Eugene&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryoftheoryo02dick_0%2Fpage%2F191&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" 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=Heronian_triangle&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Heronian_triangle"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HeronianTriangle.html">"Heronian triangle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Heronian+triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHeronianTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></span></li> <li>Online Encyclopedia of Integer Sequences <a rel="nofollow" class="external text" href="http://oeis.org/search?q=Heronian">Heronian</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._sh._Kozhegel'dinov1994" class="citation cs2">S. sh. Kozhegel'dinov (1994), "On fundamental Heronian triangles", <i>Math. Notes</i>, <b>55</b> (2): 151–6, <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%2FBF02113294">10.1007/BF02113294</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:115233024">115233024</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Notes&rft.atitle=On+fundamental+Heronian+triangles&rft.volume=55&rft.issue=2&rft.pages=151-6&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2FBF02113294&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115233024%23id-name%3DS2CID&rft.au=S.+sh.+Kozhegel%27dinov&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeronian+triangle" class="Z3988"></span></li></ul>    </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=Heronian_triangle&oldid=1238662411">https://en.wikipedia.org/w/index.php?title=Heronian_triangle&oldid=1238662411</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:Arithmetic_problems_of_plane_geometry" title="Category:Arithmetic problems of plane geometry">Arithmetic problems of plane geometry</a></li><li><a href="/wiki/Category:Types_of_triangles" title="Category:Types of triangles">Types of triangles</a></li><li><a href="/wiki/Category:Eponymous_geometric_shapes" title="Category:Eponymous geometric shapes">Eponymous geometric shapes</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><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_January_2023" title="Category:Articles with unsourced statements from January 2023">Articles with unsourced statements from January 2023</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</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 5 August 2024, at 01:28<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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Heronian triangle - Wikipedia