@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:-GKWK9C3G-P
  skos:prefLabel "géométrie euclidienne"@fr, "Euclidean geometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-JP1PHSKD-9 .

psr:-JP1PHSKD-9
  skos:prefLabel "relation d'Euler dans le triangle"@fr, "Euler's theorem in geometry"@en ;
  skos:altLabel "relations d'Euler dans le triangle"@fr ;
  dc:modified "2023-08-10"^^xsd:date ;
  skos:related psr:-RX61SX55-G ;
  skos:definition """In geometry, <b>Euler's theorem</b> states that the distance <i>d</i> between the circumcenter and incenter of a triangle is given by
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle d^{2}=R(R-2r)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>d</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>R</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>R</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>r</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle d^{2}=R(R-2r)}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f800c5d9c4acd79586d1a7482b10b5ed9f2eb9a4" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:15.76ex; height:3.176ex;" alt="{\\\\displaystyle d^{2}=R(R-2r)}"></div>
<br/>or equivalently
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\rac {1}{R-d}}+{\\rac {1}{R+d}}={\\rac {1}{r}},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi>R</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>d</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi>R</mi>
<br/>              <mo>+</mo>
<br/>              <mi>d</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mi>r</mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {1}{R-d}}+{\\rac {1}{R+d}}={\\rac {1}{r}},}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fcc54cdab2027694aba87f9806229d6bfb686d" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.171ex; width:21.897ex; height:5.509ex;" alt="{\\\\displaystyle {\\rac {1}{R-d}}+{\\rac {1}{R+d}}={\\rac {1}{r}},}"></div>
<br/>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 R}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>R</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="R"></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 r}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>r</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle r}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="r"></span> denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for  Leonhard Euler, who published it in 1765. However, the same result was published earlier by William Chapple in 1746.
<br/>From the theorem follows the <b>Euler inequality</b>:
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle R\\\\geq 2r,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>R</mi>
<br/>        <mo>≥<!-- ≥ --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>r</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R\\\\geq 2r,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632add8388428c1ed450b48bb4a24d77f77d491c" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.671ex; width:7.72ex; height:2.509ex;" alt="{\\\\displaystyle R\\\\geq 2r,}"></div>
<br/>which holds with equality only in the equilateral case. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry">https://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry</a>)"""@en, """Les relations d'Euler dans le triangle sont des relations entre les rayons des cercles inscrit/exinscrits et circonscrit. Leonhard Euler les a publiées en 1767, mais elles l'avaient déjà été par William Chappie en 1746. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Relations_d%27Euler_dans_le_triangle">https://fr.wikipedia.org/wiki/Relations_d%27Euler_dans_le_triangle</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Relations_d%27Euler_dans_le_triangle>, <https://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry> ;
  skos:broader psr:-GKWK9C3G-P ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-RX61SX55-G
  skos:prefLabel "triangle"@fr, "triangle"@en ;
  a skos:Concept ;
  skos:related psr:-JP1PHSKD-9 .