{"id":7897,"date":"2022-08-26T14:33:41","date_gmt":"2022-08-26T12:33:41","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7897"},"modified":"2022-08-26T14:33:42","modified_gmt":"2022-08-26T12:33:42","slug":"trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/08\/26\/trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale\/","title":{"rendered":"Trouver le point du cercle pour lequel la distance totale est minimale"},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"857\" height=\"649\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image.png\" alt=\"\" class=\"wp-image-7899\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image.png 857w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-300x227.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-768x582.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-600x454.png 600w\" sizes=\"auto, (max-width: 857px) 100vw, 857px\" \/><\/figure>\n\n\n\n<p>Je me trouve \u00e0 mon domicile (D), en pleine campagne o\u00f9 il n&#8217;y a pas de r\u00e9seau t\u00e9l\u00e9phonique.<\/p>\n\n\n\n<p>Je dois me rendre chez mon ami Fabien (F), isol\u00e9 lui aussi. Mais comme je suis une personne polie, je souhaite appeler Fabien afin de le pr\u00e9venir de ma venue. Je dois donc me rendre au point (P) le plus proche possible de chez moi o\u00f9 il y a du r\u00e9seau (donc sur le cercle de centre A, o\u00f9 A est l&#8217;antenne r\u00e9seau) de sorte \u00e0 minimiser la distance totale que je devrai parcourir (donc minimiser DP + PF).<\/p>\n\n\n\n<p>La question est la suivante: o\u00f9 doit se trouver le point P ?<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/08\/26\/trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale\/#La_recherche_cartesienne_est-ce_une_bonne_idee\" >La recherche cart\u00e9sienne: est-ce une bonne id\u00e9e ?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/08\/26\/trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale\/#La_recherche_avec_geogebra\" >La recherche avec g\u00e9ogebra<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/08\/26\/trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale\/#Conjecture\" >Conjecture<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_recherche_cartesienne_est-ce_une_bonne_idee\"><\/span>La recherche cart\u00e9sienne: est-ce une bonne id\u00e9e ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Je vais simplifier le sch\u00e9ma et me rapporter \u00e0 un rep\u00e8re orthonorm\u00e9 d&#8217;origine A.<\/p>\n\n\n\n<p>Dans ce rep\u00e8re, le cercle a pour \u00e9quation cart\u00e9sienne:$$x^2+y^2=r^2,$o\u00f9 \\(r\\) est le rayon du cercle. <\/p>\n\n\n\n<p>Je note maintenant:$$D(x_D;y_D)\\quad;\\quad F(x_F;y_F)$$les coordonn\u00e9es respectives de D et F.<\/p>\n\n\n\n<p>Je note \\(P(x_P;y_P)\\) tel que \\(x_P^2+y_P^2=r^2\\) puisque P appartient au cercle.<\/p>\n\n\n\n<p>On a alors:$$DP=\\sqrt{(x_P-x_D)^2 + (y_P-y_D)^2}\\quad;\\quad PF=\\sqrt{(x_P-x_F)^2 + (y_P-y_F)^2}.$$<\/p>\n\n\n\n<p>On cherche une condition sur \\(x_P\\) (par exemple) pour minimiser DP+PF. Cela n&#8217;\u00e9quivaut pas \u00e0 minimiser DP\u00b2+PF\u00b2, donc nous sommes oblig\u00e9s de garder les racines carr\u00e9es.<\/p>\n\n\n\n<p>Notons:$$f(x,y)=\\sqrt{(x-x_D)^2 + (y-y_D)^2} + \\sqrt{(x-x_F)^2 + (y-y_F)^2}$$avec x\u00b2+y\u00b2=<em>r<\/em>\u00b2. Alors,$$f(x,y)=\\sqrt{r^2+x_D^2+y_D^2-2xx_D-2yy_D} + \\sqrt{r^2+x_F^2+y_F^2-2xx_F-2yy_F}.$$<\/p>\n\n\n\n<p>Comme \\(y = \\sqrt{r^2-x^2}\\), on peut se ramener \u00e0 une fonction \u00e0 une variable:$$d(x)=f(x,y(x))=\\sqrt{r^2+x_D^2+y_D^2-2xx_D-2y_D\\sqrt{r^2-x^2}} + \\sqrt{r^2+x_F^2+y_F^2-2xx_F-2y_F\\sqrt{r^2-x^2}}.$$<\/p>\n\n\n\n<p>Sa d\u00e9riv\u00e9e est:$$d'(x)=\\frac{-x_D+\\frac{y_Dx}{\\sqrt{r^2-x^2}}}{\\sqrt{r^2+x_D^2+y_D^2-2xx_D-2y_D\\sqrt{r^2-x^2}}} + \\frac{-x_F+\\frac{y_Fx}{\\sqrt{r^2-x^2}}}{\\sqrt{r^2+x_F^2+y_F^2-2xx_F-2y_F\\sqrt{r^2-x^2}}}.$$<\/p>\n\n\n\n<p>Il faudrait trouver le signe de cette magnifique d\u00e9riv\u00e9e&#8230; mais cela semble plut\u00f4t ardu&#8230; L&#8217;approche cart\u00e9sienne ne semble pas \u00eatre la meilleure \u00e0 envisager&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_recherche_avec_geogebra\"><\/span>La recherche avec g\u00e9ogebra<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>R\u00e9alisons un sch\u00e9ma sur lequel nous allons placer un point P (que nous pourrons d\u00e9placer) sur le cercle.<\/p>\n\n\n\n<p>Tra\u00e7ons les segments DP et PF, puis cr\u00e9ons la variable <em>a<\/em> repr\u00e9sentant la somme de ces distances. Bougeons ensuite P de sorte \u00e0 rendre minimum cette somme.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"810\" height=\"508\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/geogebra-somme-distance.gif\" alt=\"\" class=\"wp-image-7905\"\/><\/figure>\n\n\n\n<p>Ma premi\u00e8re id\u00e9e \u00e9tait de supposer que P \u00e9tait sur la bissectrice de l&#8217;angle \\(\\widehat{DAF}\\), donc je m&#8217;empresse de tracer cette derni\u00e8re:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"461\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1-1024x461.png\" alt=\"\" class=\"wp-image-7906\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1-1024x461.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1-600x270.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1-300x135.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1-768x346.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-1.png 1293w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Snif! Ce n&#8217;est pas \u00e7a&#8230;<\/p>\n\n\n\n<p>J&#8217;ai donc r\u00e9fl\u00e9chir longtemps&#8230; longtemps&#8230; tr\u00e8s longtemps&#8230; une vingtaine de minutes (quand-m\u00eame!).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conjecture\"><\/span>Conjecture<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>J&#8217;ai trac\u00e9 les tangentes &#8220;sous-jacentes&#8221; au cercle passant par D et F, ai pris leur intersection I, puis ai trac\u00e9 la droite (AI). Le point d&#8217;intersection de (AI) et du cercle semble passer par le point P trouv\u00e9 pr\u00e9c\u00e9demment.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"591\" height=\"589\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-2.png\" alt=\"\" class=\"wp-image-7907\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-2.png 591w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-2-160x160.png 160w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-2-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/image-2-150x150.png 150w\" sizes=\"auto, (max-width: 591px) 100vw, 591px\" \/><\/figure>\n<\/div>\n\n\n<p>J&#8217;ai ensuite test\u00e9 plusieurs autres configurations: cela semble exact ! Il ne reste plus qu&#8217;\u00e0 le d\u00e9montrer&#8230; mais l\u00e0, je suis trop fatigu\u00e9 ! \ud83d\ude42 Je reprendrai cet article quand j&#8217;aurai trouv\u00e9 !<\/p>\n\n\n\n<p>En attendant, vous pouvez faire mumuse avec le fichier geogebra :<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-bfbef6fa-2abb-4225-9d29-9899c2c61846\" href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/somme-distance.ggb\">somme-distance<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/08\/somme-distance.ggb\" class=\"wp-block-file__button\" download aria-describedby=\"wp-block-file--media-bfbef6fa-2abb-4225-9d29-9899c2c61846\">T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Je me trouve \u00e0 mon domicile (D), en pleine campagne o\u00f9 il n&#8217;y a pas de r\u00e9seau t\u00e9l\u00e9phonique. Je dois me rendre chez mon ami Fabien (F), isol\u00e9 lui aussi. Mais comme je suis une personne polie, je souhaite appeler Fabien afin de le pr\u00e9venir de ma venue. Je dois [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":7910,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-7897","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trouver le point du cercle pour lequel la distance totale est minimale - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"\u00c9tant donn\u00e9s deux points D et F et un cercle, trouver le point P sur le cercle de sorte que la somme DP+PF soit minimale.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/08\/26\/trouver-le-point-du-cercle-pour-lequel-la-distance-totale-est-minimale\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trouver le point du cercle pour lequel la distance totale est minimale - 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