mercredi, mai 14, 2008


J'ai trouvé cette bien jolie explication de la démonstration du théorème de Pythagore par Euclide. Comment ne pas aimer la géométrie après ça?

Avant de la lire, voici le texte d'Euclide:

Dans les triangles rectangles, le carré sur le côté sous-tendant l'angle droit est égal aux carrés sur les côtés contenant l'angle droit.

Soit le triangle rectangle ABC ayant l'angle sous BAC droit. Je dis que le carré sur BC est égal aux carrés sur BA, AC.

En effet d'une part que le carré BDEC soit décrit sur BC, d'autre part les carrés GB, HC sur

BA, AC (Prop. 46) et que par le point A, soit menée AL, parallèle à l'une quelconque des BD, CE (Prop. 31). Et que AD, FC soient jointes (Dem. 1).

Puisque chacun des angles sous BAC, BAG est droit, alors relativement à une certaine droite: BA, et en un point qui est sur elle: A, les deux droites AC, AG, non placées du même côté, font des angles adjacents égaux à deux droits. Donc CA est en alignement avec AG (Prop. 14). Alors pour la même raison BA est aussi en alignement avec AH.

Et puisque l'angle sous DBC est égal à celui sous FBA car chacun est droit (Dem. 4), que celui sous ABC soit ajouté de part et d'autre: celui sous DBA tout entier est donc égal à celui sous FBC tout entier (N.C. 2).

Et puisque, d'une part DB est égale à BC, d'autre part FB à BA, alors les deux DB, BA sont égales aux deux FB, BC2, chacune à chacune et l'angle sous DBA est égal à celui sous FBC. La base AD {est} donc égale à la base FC, et le triangle ABD égal au triangle FBC (Prop. 4).

Et le parallélogramme BL {est} double du triangle ABD, car ils ont la même base BD, et sont dans les mêmes parallèles: BD, AL (Prop. 41). D'autre part le carré GB est double du triangle FBC, car, de nouveau, ils ont la même base FB, et sont dans les mêmes parallèles: FB, GC. {Or les doubles de choses égales sont égaux entre eux}3. Donc le parallélogramme BL est égal au carré GB (N.C. 1). Alors semblablement, AE et BK étant jointes, il sera démontré aussi que le parallélogramme CL est égal au carré HC. Donc le carré BDEC tout entier est égal aux deux carrés GB, HC, et, d'une part BDEC est le carré qui a été décrit sur BC, d'autre part GB et HC sont les carrés sur BA, AC. Donc le carré sur le côté BC est égal aux carrés sur les côtés BA, AC.

Donc, dans les triangles rectangles, le carré sur le côté sous-tendant l'angle droit est égal aux carrés sur les côtés contenant l'{angle} droit. Ce qu'il fallait démontrer.

28 commentaires:

Anarcho-pragmatiste a dit…

Très intéressant, même pour un mathématicien!

Normand Baillargeon a dit…


Merci du commentaire.

En plus, il me semble que c'est d'une étonnante et émouvante beauté.


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