@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-WW75956Z-P
  skos:prefLabel "right triangle"@en, "triangle rectangle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-CN7TH1V4-K .

psr:-GR8X1BV4-T
  skos:prefLabel "golden ratio"@en, "nombre d'or"@fr ;
  a skos:Concept ;
  skos:related psr:-CN7TH1V4-K .

psr:-CN7TH1V4-K
  skos:exactMatch <https://en.wikipedia.org/wiki/Kepler_triangle>, <https://fr.wikipedia.org/wiki/Triangle_de_Kepler> ;
  skos:broader psr:-WW75956Z-P ;
  dc:created "2023-07-31"^^xsd:date ;
  a skos:Concept ;
  skos:related psr:-GR8X1BV4-T ;
  skos:definition """Un <b>triangle de Kepler</b> est un triangle rectangle dont les carrés des longueurs des côtés sont en progression géométrique selon la raison du nombre d'or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varphi ={\\rac {1+{\\\\sqrt {5}}}{2}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>φ<!-- φ --></mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mn>1</mn>
<br/>              <mo>+</mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <msqrt>
<br/>                  <mn>5</mn>
<br/>                </msqrt>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mn>2</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varphi ={\\rac {1+{\\\\sqrt {5}}}{2}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b498bd7bebdaa79ba86131a9f839f96a4e7628f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:12.556ex; height:5.843ex;" alt="{\\\\displaystyle \\\\varphi ={\\rac {1+{\\\\sqrt {5}}}{2}}}"></span>. Les rapports des longueurs des côtés sont donc <span class="texhtml">1&nbsp;: <span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"><i>φ</i></span></span>&nbsp;: <i>φ</i></span> (approximativement 1&nbsp;: 1,272&nbsp;: 1,618).
<br/>Les angles non droits valent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\arcsin {1 \\\\over \\\\varphi }}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>arcsin</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mi>φ<!-- φ --></mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\arcsin {1 \\\\over \\\\varphi }}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362cc7df6e25bb10ed76ce5c4fc755799541649d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:8.705ex; height:5.676ex;" alt="{\\\\displaystyle \\\\arcsin {1 \\\\over \\\\varphi }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\arcsin {1 \\\\over {\\\\sqrt {\\\\varphi }}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>arcsin</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <msqrt>
<br/>                <mi>φ<!-- φ --></mi>
<br/>              </msqrt>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\arcsin {1 \\\\over {\\\\sqrt {\\\\varphi }}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b860ae8011224ecff0f6c8735d72d7a9d4764de" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:10.641ex; height:6.176ex;" alt="{\\\\displaystyle \\\\arcsin {1 \\\\over {\\\\sqrt {\\\\varphi }}}}"></span> radians, soit environ 38° et 52°.
<br/>Les triangles possédant de telles propriétés portent le nom du mathématicien et astronome allemand Johannes Kepler (1571-1630), qui le premier démontra que ces triangles sont caractérisés par un rapport entre le petit côté et l'hypoténuse égal au nombre d'or. Ces triangles combinent le théorème de Pythagore et le nombre d'or, notions qui fascinaient Kepler. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Triangle_de_Kepler">https://fr.wikipedia.org/wiki/Triangle_de_Kepler</a>)"""@fr, """A <b>Kepler triangle</b> is a special right triangle with edge lengths in geometric progression. The ratio of the progression 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 {\\\\sqrt {\\\\varphi }}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mi>φ<!-- φ --></mi>
<br/>          </msqrt>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\sqrt {\\\\varphi }}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1094106c2237fda1e1d4f9e6e1bde3e464637b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:3.456ex; height:3.009ex;" alt="{\\\\sqrt  \\\\varphi }"></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 \\\\varphi =(1+{\\\\sqrt {5}})/2}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>φ<!-- φ --></mi>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mn>5</mn>
<br/>          </msqrt>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo>/</mo>
<br/>        </mrow>
<br/>        <mn>2</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varphi =(1+{\\\\sqrt {5}})/2}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f409bafc8166035e6535ed6bb1a12ccb2d97d65" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.854ex; height:3.009ex;" alt="\\\\varphi =(1+{\\\\sqrt  {5}})/2"></span> is the golden ratio, and the progression can be written: <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 1:{\\\\sqrt {\\\\varphi }}:\\\\varphi }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>1</mn>
<br/>        <mo>:</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mi>φ<!-- φ --></mi>
<br/>          </msqrt>
<br/>        </mrow>
<br/>        <mo>:</mo>
<br/>        <mi>φ<!-- φ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 1:{\\\\sqrt {\\\\varphi }}:\\\\varphi }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d877fee12ff439aa86ab19fcd5744b8b630634bc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:10.013ex; height:3.009ex;" alt="1:{\\\\sqrt  \\\\varphi }:\\\\varphi "></span>,</span> or approximately <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 1:1.272:1.618}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>1</mn>
<br/>        <mo>:</mo>
<br/>        <mn>1.272</mn>
<br/>        <mo>:</mo>
<br/>        <mn>1.618</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 1:1.272:1.618}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95de289dabf5e9b1d358d45a70ba5ff2a4495675" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:15.63ex; height:2.176ex;" alt="{\\\\displaystyle 1:1.272:1.618}"></span></span>. Squares on the edges of this triangle have areas in another geometric progression, <span class="mwe-math-element"><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:\\\\varphi :\\\\varphi ^{2}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>1</mn>
<br/>        <mo>:</mo>
<br/>        <mi>φ<!-- φ --></mi>
<br/>        <mo>:</mo>
<br/>        <msup>
<br/>          <mi>φ<!-- φ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 1:\\\\varphi :\\\\varphi ^{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2539bead36ab49e9de5a5e623a6dd50fde5c637" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\\\\displaystyle 1:\\\\varphi :\\\\varphi ^{2}}"></span>. Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles.
<br/>This triangle is named after Johannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Kepler_triangle">https://en.wikipedia.org/wiki/Kepler_triangle</a>)"""@en ;
  skos:prefLabel "triangle de Kepler"@fr, "Kepler triangle"@en ;
  skos:inScheme psr: ;
  dc:modified "2023-07-31"^^xsd:date .