{"id":294,"date":"2018-06-09T09:44:57","date_gmt":"2018-06-09T07:44:57","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=294"},"modified":"2018-08-12T06:58:26","modified_gmt":"2018-08-12T04:58:26","slug":"sangaku-celebre-de-hidetoshi-fukagawa","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/","title":{"rendered":"Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa"},"content":{"rendered":"

Les sangakus<\/em> japonaises sont de petits chefs d\u2019\u0153uvres aussi bien au niveau du raisonnement math\u00e9matique que de l\u2019esth\u00e9tique. Jean Constant, par exemple, s\u2019en est fait une sp\u00e9cialit\u00e9 (voir l\u2019image mise en avant). La sangaku suivante a \u00e9t\u00e9 d\u00e9couverte par Hidetoshi Fukagawa.<\/p>\n

\"\"Les deux triangles (rouge et vert) inscrits dans le carr\u00e9 jaune sont \u00e9quilat\u00e9raux, quel est le rapport entre les rayons des cercles bleus ?<\/p>\n

Rayon d’un cercle inscrit<\/h2>\n

Les deux cercles sont inscrits dans deux triangles. Un th\u00e9or\u00e8me permet d’en calculer les rayons en fonction de leurs aires et de leurs p\u00e9rim\u00e8tres. Plus pr\u00e9cis\u00e9ment, le rayon du cercle inscrit dans un triangle est \u00e9gal \u00e0 deux fois la surface du triangle divis\u00e9 par son p\u00e9rim\u00e8tre, ce r\u00e9sultat est mis en \u00e9vidence par un dessin\u00a0: l\u2019aire du triangle se d\u00e9compose en\u00a0 trois triangles de m\u00eame hauteur, le rayon du cercle inscrit. L\u2019aire de chacun de ces triangles est donc \u00e9gale au rayon du cercle inscrit multipli\u00e9 par la longueur du c\u00f4t\u00e9 oppos\u00e9 divis\u00e9e par deux. En faisant la somme, le p\u00e9rim\u00e8tre du triangle s\u2019introduit naturellement<\/span> .<\/span><\/p>\n

\"\"<\/p>\n

Plan d’attaque du probl\u00e8me<\/h2>\n

Pour calculer les rayons des deux cercles, il s\u2019agit donc de calculer un certain nombre de longueurs de segments de la figure. L\u2019id\u00e9e pour les calculer vient si nous en oublions une partie. En utilisant les angles de 60\u00b0 et de 45\u00b0 en \u00e9vidence, nous trouvons que les triangles rouges ont les m\u00eames angles et sont donc semblables. <\/span><\/span><\/p>\n

\"\"<\/p>\n

Gr\u00e2ce aux rapports de similitude et au th\u00e9or\u00e8me de Pythagore, les mesures de longueurs apparaissent progressivement, une d\u2019entre elles (AC) ayant \u00e9t\u00e9 choisie comme unit\u00e9. Le dessin est utile pour suivre le raisonnement.<\/span> Nous en d\u00e9duisons progressivement les diverses longueurs importantes. Elles sont not\u00e9es sur le dessin ci-dessous.<\/span><\/span><\/p>\n

\"\"<\/p>\n

On en d\u00e9duit les valeurs des deux rayons\u00a0:<\/p>\n

\"\"Un calcul alg\u00e9brique<\/h2>\n

Un calcul alg\u00e9brique permet de montrer que R\u00a0=\u00a02\u00a0r<\/em>. Pour cette derni\u00e8re \u00e9tape, aucune visualisation n\u2019est n\u00e9cessaire et nous pouvons l\u2019ex\u00e9cuter avec un logiciel de calcul formel. Ce dernier calcul nous entra\u00eene vers les extensions alg\u00e9briques, nous nous arr\u00eaterons \u00e0 leur porte.<\/p>\n

L’\u00e9ventail de la geisha<\/h2>\n

Dans certaines sangakus<\/em>, les auteurs ont clairement privil\u00e9gi\u00e9 l\u2019esth\u00e9tique.<\/p>\n

\"\"<\/p>\n

Par exemple, dans celui en forme d\u2019\u00e9ventail ouvert aux deux tiers ci-dessus, il s\u2019agit de trouver le rapport entre les rayons des cercles verts et rouges. Ici encore, l\u2019essentiel est d\u2019introduire les bons points, qui ne sont pas directement visibles. On trouve :<\/p>\n

\"\"<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Les sangakus japonaises sont de petits chefs d\u2019\u0153uvres aussi bien au niveau du raisonnement math\u00e9matique que de l\u2019esth\u00e9tique. Jean Constant, par exemple, s\u2019en est fait une sp\u00e9cialit\u00e9 (voir l\u2019image mise en avant). La sangaku suivante a \u00e9t\u00e9 d\u00e9couverte par Hidetoshi Fukagawa. Les deux triangles (rouge et vert) inscrits dans le carr\u00e9 jaune sont \u00e9quilat\u00e9raux, quel … Continuer la lecture de Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":314,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[5,8,12,29,10],"tags":[49,139,140,40],"class_list":["post-294","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-arts","category-enigmes","category-graphismes","category-maths-2","category-voyages","tag-enigmes","tag-geometrie","tag-japon","tag-mathematiques"],"yoast_head":"\nUne sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa, 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\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Les sangakus japonaises sont de petits chefs d\u2019\u0153uvres aussi bien au niveau du raisonnement math\u00e9matique que de l\u2019esth\u00e9tique. Jean Constant, par exemple, s\u2019en est fait une sp\u00e9cialit\u00e9 (voir l\u2019image mise en avant). La sangaku suivante a \u00e9t\u00e9 d\u00e9couverte par Hidetoshi Fukagawa. Les deux triangles (rouge et vert) inscrits dans le carr\u00e9 jaune sont \u00e9quilat\u00e9raux, quel … Continuer la lecture de Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa →\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/\" \/>\n<meta property=\"og:site_name\" content=\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\" \/>\n<meta property=\"article:published_time\" content=\"2018-06-09T07:44:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2018-08-12T04:58:26+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/06\/VisuCom7.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1181\" \/>\n\t<meta property=\"og:image:height\" content=\"1181\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/\",\"name\":\"Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa, par Herv\u00e9 Lehning\",\"isPartOf\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/06\/VisuCom7.png\",\"datePublished\":\"2018-06-09T07:44:57+00:00\",\"dateModified\":\"2018-08-12T04:58:26+00:00\",\"author\":{\"@id\":\"\"},\"breadcrumb\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/#primaryimage\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/06\/VisuCom7.png\",\"contentUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/06\/VisuCom7.png\",\"width\":1181,\"height\":1181},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/06\/09\/sangaku-celebre-de-hidetoshi-fukagawa\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Une sangaku c\u00e9l\u00e8bre, de Hidetoshi Fukagawa\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\",\"name\":\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\",\"description\":\"Comment comprendre le monde moderne sans culture math\u00e9matique ? 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