Exercice de construction
dans Géométrie
Bonjour,
Je vous propose ce (petit) exercice.
Construire un cercle passant par un point donné $A$ et tangent à deux cercles $O$ et $O'$.
Je vous propose ce (petit) exercice.
Construire un cercle passant par un point donné $A$ et tangent à deux cercles $O$ et $O'$.
Connectez-vous ou Inscrivez-vous pour répondre.
Réponses
<applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar"
codebase="http://jars.geogebra.org/webstart/4.2/unsigned/"
width="800" height="400">
<param name="ggbBase64" value="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" />
<param name="java_arguments" value="-Xmx1024m -Djnlp.packEnabled=true" />
<param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_algos.jar, geogebra_export.jar, geogebra_javascript.jar, jlatexmath.jar, jlm_greek.jar, jlm_cyrillic.jar, geogebra_properties.jar" />
<param name="cache_version" value="4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0" />
<param name="showResetIcon" value="false" />
<param name="enableRightClick" value="false" />
<param name="errorDialogsActive" value="true" />
<param name="enableLabelDrags" value="false" />
<param name="showMenuBar" value="false" />
<param name="showToolBar" value="false" />
<param name="showToolBarHelp" value="false" />
<param name="showAlgebraInput" value="false" />
<param name="useBrowserForJS" value="true" />
<param name="allowRescaling" value="true" />
C'est une appliquette Java créée avec GeoGebra ( www.geogebra.org) - Il semble que Java ne soit pas installé sur votre ordinateur, merci d'aller sur www.java.com
</applet>
Bonne journée grise.
RC
@Philippe Malot : lorsqu'une inversion de pôle $A$ transforme les deux cercles en deux cercles de même rayon.
voici ue figure montrant les quatre solutions
$B,B^{\prime }$ sont les images de A par les inversions échangeant les deux cercles et ayant pour pôles leurs centres d'homothétie $H$ et $H^{\prime }$.
Il ne reste plus qu'à construire les deux cercles tangents à $\left( O\right) $ passant par $A$ et $B$ et les deux cercles tangents à $\left( O\right) $ passant par $A$ et $B^{\prime }$.
Pour construire les deux cercles tangents à $\left( O\right) $ passant par $A$ et $B$, remarquer que l'axe radical de $\left( O\right) $ et d'un cercle variable passant par $A$ et $B$ passe par un point fixe $F$ de la droite $AB$; les points de contact avec $\left( O\right) $ des deux cercles cherchés sont les points de contact des tangentes menées de $F$ à $\left( O\right) $.
En fait, ce problème est un cas particulier du fameux problème d'Apollonius qui consiste à construire les huit cercles tangents à trois cercles donnés.
Merci pour contribution, effectivement je n'avais pas pensé à vérifier dans le Lebossé-Hémery.
Je l'ai pris dans un livre d'exercices corrigés de Math Elém (Ligel). Les auteurs ne font pas appel à l'inversion, mais c'est sous-jacent bien sûr.
Je rajoute l'applet avec les quatre cercles (dans les bons cas).
<applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar"
codebase="http://jars.geogebra.org/webstart/4.2/unsigned/"
width="600" height="400">
<param name="ggbBase64" 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/>
<param name="java_arguments" value="-Xmx1024m -Djnlp.packEnabled=true" />
<param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_algos.jar, geogebra_export.jar, geogebra_javascript.jar, jlatexmath.jar, jlm_greek.jar, jlm_cyrillic.jar, geogebra_properties.jar" />
<param name="cache_version" value="4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0, 4.2.10.0" />
<param name="showResetIcon" value="false" />
<param name="enableRightClick" value="false" />
<param name="errorDialogsActive" value="true" />
<param name="enableLabelDrags" value="false" />
<param name="showMenuBar" value="false" />
<param name="showToolBar" value="false" />
<param name="showToolBarHelp" value="false" />
<param name="showAlgebraInput" value="false" />
<param name="useBrowserForJS" value="true" />
<param name="allowRescaling" value="true" />
C'est une appliquette Java créée avec GeoGebra ( www.geogebra.org) - Il semble que Java ne soit pas installé sur votre ordinateur, merci d'aller sur www.java.com
</applet>
Le lemme suivant (dont j'ai changé la formulation) est la base de la construction trouvée par Gergonne des cercles tangents à trois cercles donnés $\left( O\right) ,\left( O^{\prime }\right) ,\left( O^{\prime \prime }\right) $ :
Soient
$H_{1}$ un des deux centres d'homothétie de $\left( O\right) $ et $\left( O^{\prime }\right) $
$H_{2}$ un des deux centres d'homothétie de $\left( O\right) $ et $\left( O^{\prime \prime }\right) $
$M$ le pôle de la droite $H_{1}H_{2}$ par rapport à $\left( O\right) $
$N$ le point d'intersection de la polaire de $H_{1}$ par rapport à $\left( O^{\prime }\right) $ et de celle de $H_{2}$ par rapport à $\left( O^{\prime \prime }\right) $
Alors chacun des points d'intersection $P$ et $Q$ du cercle $\left( O\right) $ et de la droite $MN$ est point de contact avec $\left( O\right) $ d'un cercle tangent aux trois cercles donnés
Alors un petit exercice : prouver ce lemme (dont je n'ai pas trouvé la démonstration originale) ?
Ci-dessous une figure et l'énoncé original de Gergonne (j'ai simplement remplacé la corde de contact des tangentes issues d'un point à un cercle par la polaire du point par rapport au cercle) et ne me suis pas limité aux centres extérieurs d'homothétie
Cordialement. Poulbot
en fait, la preuve originale de Gergonne figure bien sur le net à
GERGONNE mais elle est analytique avec beaucoup de texte et des petits problèmes de signes résolus par des explications pas toujours très claires.
Cordialement. Pulbot
Voici une démonstration synthétique :
http://archive.numdam.org/ARCHIVE/AMPA/AMPA_1822-1823__13_/AMPA_1822-1823__13__193_0/AMPA_1822-1823__13__193_0.pdf