{"id":1272,"date":"2020-04-19T11:11:07","date_gmt":"2020-04-19T09:11:07","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=1272"},"modified":"2020-05-08T17:23:34","modified_gmt":"2020-05-08T15:23:34","slug":"le-cercle-de-conway","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2020\/04\/19\/le-cercle-de-conway\/","title":{"rendered":"Le cercle de Conway"},"content":{"rendered":"

John Horton Conway, n\u00e9 le 26 d\u00e9cembre 1937 et mort lors de la pand\u00e9mie de Coronavirus le 11\u00a0 avril 2020, est l’un des math\u00e9maticiens les plus originaux du XX\u00b0 si\u00e8cle. Il est particuli\u00e8rement connu pour sa cr\u00e9ation du jeu de la vie, qui est d\u00e9j\u00e0 l’objet d’un article de ce blog.<\/p>\n

Un cercle dans un triangle<\/h2>\n

En pleine \u00e9poque des maths modernes, le malicieux Conway a d\u00e9couvert une propri\u00e9t\u00e9 du triangle qui aurait pu l’\u00eatre par Euclide, trois si\u00e8cles avant notre \u00e8re.<\/p>\n

\"\"<\/p>\n

Soit ABC un triangle,\u00a0a<\/em>,\u00a0b<\/em> et\u00a0c <\/em>les longueurs des c\u00f4t\u00e9s BC, CA et AB. On prolonge les c\u00f4t\u00e9s comme indiqu\u00e9 sur la figure, on obtient ainsi six points not\u00e9s de petits ronds sur la figure. Conway a d\u00e9couvert que ces six points appartenaient \u00e0 un m\u00eame cercle, appel\u00e9 depuis cercle de Conway du triangle ABC en son honneur.<\/p>\n

Analyse : centre du cercle de Conway<\/h2>\n

\"\"<\/p>\n

En supposant que ce cercle existe, on d\u00e9montre en consid\u00e9rant les couples de points venant du m\u00eame sommet que son centre appartient \u00e0 chacune des bissectrices du triangle ABC. Il s’agit donc du centre du cercle inscrit I.<\/p>\n

Synth\u00e8se : cercle de Conway<\/h2>\n

\"\"<\/p>\n

On consid\u00e8re les bissectrices PP’, QQ’ et RR’ du triangle ABC et le cercle de centre I passant par l’un des points verts. On d\u00e9montre de proche en proche qu’il passe par tous les points verts puisque les droites PP’, QQ’ et RR’ sont les m\u00e9diatrices des couples de points verts contigus.<\/p>\n","protected":false},"excerpt":{"rendered":"

John Horton Conway, n\u00e9 le 26 d\u00e9cembre 1937 et mort lors de la pand\u00e9mie de Coronavirus le 11\u00a0 avril 2020, est l’un des math\u00e9maticiens les plus originaux du XX\u00b0 si\u00e8cle. Il est particuli\u00e8rement connu pour sa cr\u00e9ation du jeu de la vie, qui est d\u00e9j\u00e0 l’objet d’un article de ce blog. Un cercle dans un … Continuer la lecture de Le cercle de Conway<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":1273,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[63,29,3],"tags":[416,453,451,452,456,455,454],"class_list":["post-1272","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-histoire","category-maths-2","category-sciences","tag-bourbaki","tag-cercle","tag-conway","tag-geometrie-elementaire","tag-maths-modernes","tag-points-cocycliques","tag-victime-du-covid-19"],"yoast_head":"\nLe cercle de Conway, par Herv\u00e9 Lehning<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2020\/04\/19\/le-cercle-de-conway\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le cercle de Conway, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"John Horton Conway, n\u00e9 le 26 d\u00e9cembre 1937 et mort lors de la pand\u00e9mie de Coronavirus le 11\u00a0 avril 2020, est l’un des math\u00e9maticiens les plus originaux du XX\u00b0 si\u00e8cle. Il est particuli\u00e8rement connu pour sa cr\u00e9ation du jeu de la vie, qui est d\u00e9j\u00e0 l’objet d’un article de ce blog. 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