LE THÉORÈME DE CÉVA.

Ci dessous , vous trouverez différentes façons de démontrer le théorème de Céva ainsi que quelques applications évidentes

Le théorème de Céva :

a ,b et c étant trois points de (BC), (AC) et (AB), les droites (Aa) , (Bb) et (Cc) sont concourantes si et seulement si \dfrac{\overline{aB}}{\overline{aC}}\times\dfrac{\overline{bC}}{\overline{bA}}\times\dfrac{\overline{cA}}{\overline{cB}}=-1

Rendered by QuickLaTeX.com

La première démonstration utilise le « théorème du chevron »

\bullet Soit ABC un triangle, Q un point de (CB) et P un point de (AQ), alors \dfrac{\overline{QC}}{\overline{QB}}=\dfrac{\mathscr{A}(QCA)}{\mathscr{A}(QBA)}…. =\dfrac{\mathscr{A}(QCP)}{\mathscr{A}(QBP)} pour les mêmes raisons… =\dfrac{\mathscr{A}(PCA)}{\mathscr{A}(PBA)} par différence (chevron)
On utilise le fait que ces triangles ont le même hauteur.

Rendered by QuickLaTeX.com

Le théorème se démontre alors facilement :

En effet :\dfrac{\overline{aB}}{\overline{aC}}=\dfrac{\mathscr{A}(OBA)}{\mathscr{A}(OCA)} , \dfrac{\overline{bC}}{\overline{bA}}=\dfrac{\mathscr{A}(OCB)}{\mathscr{A}(OAB)} et \dfrac{\overline{cA}}{\overline{cB}}=\dfrac{\mathscr{A}(OAC)}{\mathscr{A}(OBC)} ( chevron et rechevron et ..)

La loi des sinus permet aussi de démontrer le théorème de Céva :

Soit ABC un triangle et O le centre de son cercle circonscrit de rayon R, D le point diamétralement opposé à B.

Rendered by QuickLaTeX.com

(\overrightarrow{DC},\overrightarrow{DB})=(\overrightarrow{AC},\overrightarrow{AB}) et donc BC=2R \sin \hat{A}

De même que AC=2R \sin \hat{B} et AB=2R \sin \hat{C}.

On en déduit \dfrac{BC}{\sin\hat{A}}=\dfrac{AC}{\sin\hat{B}}=\dfrac{AB}{\sin\hat{C}}

Le théorème se démontre alors ainsi:

Il suffit de considérer les trois triangles OAB, BOC et COA qui ont pour sommet O.

\dfrac{OB}{OA}=\dfrac{\sin\hat{OAB}}{\sin\hat{OBA}} , \dfrac{OC}{OB}=\dfrac{\sin\hat{OBC}}{\sin\hat{OCB}} , \dfrac{OA}{OC}=\dfrac{\sin\hat{OCA}}{\sin\hat{OAC}}
ce qui donne:

\dfrac{\sin\hat{OAB}}{\sin\hat{OBA}}.\dfrac{\sin\hat{OBC}}{\sin\hat{OCB}}.\dfrac{\sin\hat{OCA}}{\sin\hat{OAC}}=1

soit \dfrac{\sin\hat{aAB}}{\sin\hat{bBA}}.\dfrac{\sin\hat{bBC}}{\sin\hat{cCB}}.\dfrac{\sin\hat{cCA}}{\sin\hat{aAC}}=-1 en tenant compte maintenant de l’orientation des angles.

Puis comme \dfrac{aB}{aC}=\dfrac{\sin\hat{aAB}}{\sin\hat{aAC}}.\dfrac{AB}{AC} et idem pour \dfrac{bC}{bA} et \dfrac{cA}{cB} le résultat s’ensuit.

Premières applications:

Soit ABC un triangle quelconque, les hauteurs issues de A, B et C sont concourantes .

Rendered by QuickLaTeX.com

En effet aB=AB\times \cos \widehat{B}, aC=AC\times \cos \widehat{C}, cB=CB\times \cos \widehat{B}
Et finalement, on trouve bien \dfrac{\overline{aB}}{\overline{aC}}\times\dfrac{\overline{bC}}{\overline{bA}}\times\dfrac{\overline{cA}}{\overline{cB}}=-1

Pour les médianes, leur concours est évident car \dfrac{\overline{aB}}{\overline{aC}}=-1=\dfrac{\overline{bC}}{\overline{bA}}=\dfrac{\overline{cA}}{\overline{cB}}

Rendered by QuickLaTeX.com

On remarquera cependant que le théorème du chevron énoncé plus haut permet de démontrer que les médianes partagent le triangle en six triangles de même aire, donc
\mathscr{A}(AGB)=2\mathscr{A}(aBG) et ce même théorème permet de conclure que AG=2Ga.

Pour les bissectrices d’un triangle,on utilisera cette propriété très pratique, conséquence de la loi des sinus, à savoir qu’une bissectrice intérieure de l’angle d’un triangle divise le côté opposé proportionnellement aux longueurs des côtés adjacents.

Rendered by QuickLaTeX.com

\dfrac{Ca}{aB}=\dfrac{AC}{AB} et on obtient alors rapidement \dfrac{\overline{aB}}{\overline{aC}}\times\dfrac{\overline{bC}}{\overline{bA}}\times\dfrac{\overline{cA}}{\overline{cB}}=-1

Une autre façon de démontrer le théorème de Céva.

Notons f, g eth les équations respectives des droites (Aa) , (Bb) et (Cc)

La relation essentielle :[A,B, C] [f, g, h] = f(A)g(B)h(C) + f(B)g(C)h(A) + f(C)g(A)h(B)
-f(C)g(B)h(A) - f(B)g(A)h(C) - f(A)g(C)h(B) permet de démontrer directement le théorème, en effet :

Si les droites (Aa) , (Bb) et (Cc) sont concourantes alors [f, g, h]=0, comme de plus f(A)=g(B)=h(C)=0 alors
f(B)g(C)h(A) + f(C)g(A)h(B)=0
soit \dfrac{f(B)g(C)h(A) }{f(C)g(A)h(B)}=-1 (*)
Mais rappelons qu’en géométrie projective, si D est une droite affine d’équation f et soient a, b deux
points distincts du plan , on suppose que D coupe (ab) en m, alors on a la
propriété suivante :

\dfrac{\overline{ma}}{\overline{mb}}=\dfrac{f(a)T(b)}{f(b)T(a)}T est l’équation de {D_{\infty} .

donc
\dfrac{\overline{aB}}{\overline{aC}}\times\dfrac{\overline{bC}}{\overline{bA}}\times\dfrac{\overline{cA}}{\overline{cB}}=\dfrac{f(B)T(C)}{f(C)T(B)}\times\dfrac{g(C)T(A)}{g(A)T(C)}\times\dfrac{h(A)T(B)}{h(B)T(A)}=
\dfrac{f(B)}{f(C)}\times\dfrac{g(C)}{g(A)}\times\dfrac{h(A))}{h(B)}=-1 d’après (*)

Le théorème de Céva se démontre aussi à l’aide du théorème de Ménélaüs, en effet:

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