Comme je vous l’avais précisé dans un article précédent, nous pouvons dans le prolongement de la construction de la droite d’Euler, construire le cercle d’Euler, pour cela nous pouvons reprendre la précédente construction.

Il suffit de construire les points suivants :

  • les milieux des côtés du triangle
  • les pieds des hauteurs
  • les milieux des segments reliant les sommets à l’orthocentre

Ainsi nous pouvons constater que ces neufs points appartiennent à la circonférence d’un même cercle appelé cercle d’Euler.

Mais comment tracer un cercle alors que nous n’avons pas de centre ?

Il suffit simplement de tracer deux cordes. Mais qu’est-ce qu’une corde ?

En géométrie, une corde est un segment reliant deux points d’un cercle ou d’une autre courbe.

Puis comme vous savez tracer des médiatrices, il suffit de tracer les médiatrices de ces deux cordes. Le point d’intersection entre ces deux médiatrices vous donnera alors le centre du cercle d’Euler qu’il sera possible de construire ! 😊

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<title>GeoGebra - Protocole de construction</title>
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<body>
<table border="1">
<tr>
<th>Nom</th>
<th>Définition</th>
<th>Valeur</th>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#1B1BCD">Point A</span></td>
<td> </td>
<td><span style="color:#1B1BCD">A = (-1.23, 0.44)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#1B1BCD">Point B</span></td>
<td> </td>
<td><span style="color:#1B1BCD">B = (2.77, 5.44)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#1B1BCD">Point C</span></td>
<td> </td>
<td><span style="color:#1B1BCD">C = (10.77, 2.44)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#993300">Triangle t1</span></td>
<td><span style="color:#993300">Polygone(A, B, C)</span></td>
<td><span style="color:#993300">t1 = 26</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#993300">Segment c</span></td>
<td><span style="color:#993300">Segment(A, B, t1)</span></td>
<td><span style="color:#993300">c = 6.4</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#993300">Segment a</span></td>
<td><span style="color:#993300">Segment(B, C, t1)</span></td>
<td><span style="color:#993300">a = 8.54</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#993300">Segment b</span></td>
<td><span style="color:#993300">Segment(C, A, t1)</span></td>
<td><span style="color:#993300">b = 12.17</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#DC0000">Droite f</span></td>
<td><span style="color:#DC0000">Médiatrice(A, B)</span></td>
<td><span style="color:#DC0000">f: -4x - 5y = -17.79</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#DC0000">Droite g</span></td>
<td><span style="color:#DC0000">Médiatrice(A, C)</span></td>
<td><span style="color:#DC0000">g: -12x - 2y = -60.1</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#DC0000">Droite h</span></td>
<td><span style="color:#DC0000">Médiatrice(C, B)</span></td>
<td><span style="color:#DC0000">h: 8x - 3y = 42.31</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point CentreDuCercleCirconscrit</span></td>
<td><span style="color:#444444">Intersection(f, g)</span></td>
<td><span style="color:#444444">CentreDuCercleCirconscrit = (5.09, -0.52)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#1B1BCD">Point E</span></td>
<td> </td>
<td><span style="color:#1B1BCD">E = (10.05, 6.26)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point F</span></td>
<td><span style="color:#444444">MilieuCentre(B, A)</span></td>
<td><span style="color:#444444">F = (0.77, 2.94)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point G</span></td>
<td><span style="color:#444444">MilieuCentre(B, C)</span></td>
<td><span style="color:#444444">G = (6.77, 3.94)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point H</span></td>
<td><span style="color:#444444">MilieuCentre(A, C)</span></td>
<td><span style="color:#444444">H = (4.77, 1.44)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#0000FF">Demi-droite i</span></td>
<td><span style="color:#0000FF">DemiDroite(B, H)</span></td>
<td><span style="color:#0000FF">i: 4x + 2y = 21.96</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#0000FF">Demi-droite j</span></td>
<td><span style="color:#0000FF">DemiDroite(C, F)</span></td>
<td><span style="color:#0000FF">j: -0.5x - 10y = -29.83</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#0000FF">Demi-droite k</span></td>
<td><span style="color:#0000FF">DemiDroite(A, G)</span></td>
<td><span style="color:#0000FF">k: -3.5x + 8y = 7.87</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point CentreDeGravit鼯span></td>
<td><span style="color:#444444">Intersection(i, j)</span></td>
<td><span style="color:#444444">CentreDeGravit頽 (4.1, 2.78)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#00FF00">Droite l</span></td>
<td><span style="color:#00FF00">Perpendiculaire(B, b)</span></td>
<td><span style="color:#00FF00">l: 12x + 2y = 44.1</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#00FF00">Droite m</span></td>
<td><span style="color:#00FF00">Perpendiculaire(C, c)</span></td>
<td><span style="color:#00FF00">m: -4x - 5y = -55.29</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#00FF00">Droite n</span></td>
<td><span style="color:#00FF00">Perpendiculaire(A, a)</span></td>
<td><span style="color:#00FF00">n: -8x + 3y = 11.19</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point Orthocentre</span></td>
<td><span style="color:#444444">Intersection(l, m)</span></td>
<td><span style="color:#444444">Orthocentre = (2.11, 9.37)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td>Droite DroiteEuler</td>
<td>Droite(Orthocentre, CentreDuCercleCirconscrit)</td>
<td>DroiteEuler: 9.88x + 2.98y = 48.82</td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#DC0000">Cercle d</span></td>
<td><span style="color:#DC0000">Cercle(CentreDuCercleCirconscrit, C)</span></td>
<td><span style="color:#DC0000">d: (x - 5.09)² + (y + 0.52)² = 40.95</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#7D3E49">Demi-droite p</span></td>
<td><span style="color:#7D3E49">DemiDroite(C, B)</span></td>
<td><span style="color:#7D3E49">p: -3x - 8y = -51.86</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#7D3E49">Demi-droite q</span></td>
<td><span style="color:#7D3E49">DemiDroite(A, B)</span></td>
<td><span style="color:#7D3E49">q: -5x + 4y = 7.94</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point D</span></td>
<td><span style="color:#444444">Intersection(n, p)</span></td>
<td><span style="color:#444444">D = (0.9, 6.14)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point I</span></td>
<td><span style="color:#444444">Intersection(m, q)</span></td>
<td><span style="color:#444444">I = (4.43, 7.52)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#872154">Droite r</span></td>
<td><span style="color:#872154">Médiatrice(F, D)</span></td>
<td><span style="color:#872154">r: -0.14x - 3.2y = -14.65</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#872154">Droite s</span></td>
<td><span style="color:#872154">Médiatrice(I, G)</span></td>
<td><span style="color:#872154">s: -2.34x + 3.57y = 7.37</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#003399">Point CentreDuCercleEuler</span></td>
<td><span style="color:#003399">Intersection(DroiteEuler, r)</span></td>
<td><span style="color:#003399">CentreDuCercleEuler = (3.6, 4.43)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td>Cercle e</td>
<td>Cercle(CentreDuCercleEuler, D)</td>
<td>e: (x - 3.6)² + (y - 4.43)² = 10.24</td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point K</span></td>
<td><span style="color:#444444">Intersection(e, b, 2)</span></td>
<td><span style="color:#444444">K = (3.47, 1.23)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point L</span></td>
<td><span style="color:#444444">MilieuCentre(Orthocentre, B)</span></td>
<td><span style="color:#444444">L = (2.44, 7.41)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point M</span></td>
<td><span style="color:#444444">MilieuCentre(Orthocentre, C)</span></td>
<td><span style="color:#444444">M = (6.44, 5.91)</span></td>
</tr>
<tr style='vertical-align:baseline;'>
<td><span style="color:#444444">Point N</span></td>
<td><span style="color:#444444">MilieuCentre(Orthocentre, A)</span></td>
<td><span style="color:#444444">N = (0.44, 4.91)</span></td>
</tr>
</table>
Created with <a href="https://www.geogebra.org/" target="_blank" >GeoGebra</a>
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