{"id":7809,"date":"2022-07-10T17:38:22","date_gmt":"2022-07-10T15:38:22","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7809"},"modified":"2022-07-10T17:42:20","modified_gmt":"2022-07-10T15:42:20","slug":"trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/","title":{"rendered":"Trouver la distance entre les centres de deux cercles dans un parall\u00e9logramme"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Dans cet article, je fais suite \u00e0 <a href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/09\/trouver-la-distance-entre-deux-centres-de-cercles\/\" target=\"_blank\" rel=\"noreferrer noopener\">celui publi\u00e9 hier<\/a>, et le g\u00e9n\u00e9ralise dans le parall\u00e9logramme ci-dessous.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"723\" height=\"227\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-8.png\" alt=\"distance centres cercles parall\u00e9logramme\" class=\"wp-image-7810\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-8.png 723w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-8-300x94.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-8-600x188.png 600w\" sizes=\"auto, (max-width: 723px) 100vw, 723px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Dans cette figure, les cercles trac\u00e9s sont tangents aux c\u00f4t\u00e9s du parall\u00e9logramme et \u00e0 la diagonale trac\u00e9e.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Tout est parti du fait que je m&#8217;ennuyais en ce dimanche apr\u00e8s-midi&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 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\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#Distance_entre_les_centres_des_cercles_dans_le_parallelogramme_preliminaires\" >Distance entre les centres des cercles dans le parall\u00e9logramme: pr\u00e9liminaires<\/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\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#Le_repere\" >Le rep\u00e8re<\/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\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#Le_point_E\" >Le point E<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#Bissectrice_de_AC_et_AD\" >Bissectrice de (AC) et (AD)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#Bissectrice_de_AC_et_DC\" >Bissectrice de (AC) et (DC)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/10\/trouver-la-distance-entre-les-centres-de-deux-cercles-dans-un-parallelogramme\/#La_longueur_EF\" >La longueur EF<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Distance_entre_les_centres_des_cercles_dans_le_parallelogramme_preliminaires\"><\/span>Distance entre les centres des cercles dans le parall\u00e9logramme: pr\u00e9liminaires<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Il faut avant tout comprendre comment ces cercles ont \u00e9t\u00e9 trac\u00e9s.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Un cercle est tangent aux trois c\u00f4t\u00e9s du triangle dans lequel il est inscrit si son centre est l&#8217;intersection des bissectrices de ce triangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, le probl\u00e8me se ram\u00e8ne \u00e0 trouver la position de ces points de concours des bissectrices. Il faut donc, par exemple, d\u00e9terminer une \u00e9quation des bissectrices dans un rep\u00e8re donn\u00e9&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_repere\"><\/span>Le rep\u00e8re<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Pour y voir plus clair, on va se placer dans le rep\u00e8re d&#8217;origine A, d&#8217;axes (AB) et la droite perpendiculaire \u00e0 (AB) passant par A (non dessin\u00e9e car peu utile) et d&#8217;unit\u00e9 1 carreau.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"792\" height=\"292\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-9.png\" alt=\"\" class=\"wp-image-7811\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-9.png 792w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-9-300x111.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-9-600x221.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-9-768x283.png 768w\" sizes=\"auto, (max-width: 792px) 100vw, 792px\" \/><\/figure>\n<\/div>\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{standalone}\n\\usepackage{tikz}\n\\setlength{\\parindent}{0pt}\n\\begin{document}\n\n\\begin{tikzpicture}\n\\draw[gray,dashed] (0,0) grid (10,3);\n\\draw (0,0) node[below left] {A} -- (8,0) node[below right] {B} -- (10,3) node[above right] {C} -- (2,3) node[above left] {D} -- cycle;\n\\draw  (0,0)-- (10,3);\n\\draw  (7.41737761672328,1.0886398812302285) circle (1.0886398812302285cm);\n\\draw  (2.58262238327672,1.9113601187697722) circle (1.0886398812302278cm);\n\\draw [red] (2.58262238327672,1.9113601187697722) node[below] {E} -- (7.41737761672328,1.0886398812302285) node[below] {F} node[midway,above] {?};\n\\end{tikzpicture}\n\n\\end{document}<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_point_E\"><\/span>Le point E<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Bissectrice_de_AC_et_AD\"><\/span>Bissectrice de (AC) et (AD)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Dans ce rep\u00e8re, les \u00e9quations cart\u00e9siennes de (AC) et (AD) sont respectivement:$$(AC)\\;:\\;-3x+10y=0\\qquad;\\qquad (AD)\\;:\\;-\\frac{3}{2}x+y=0.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La distance de E \u00e0 (AC) est \u00e9gale \u00e0 celle de E \u00e0 (AD) car E est sur la bissectrice de l&#8217;angle form\u00e9 par ces deux droites, ce qui se traduit par l&#8217;\u00e9galit\u00e9:$$\\frac{|-3x_E+10y_E|}{\\sqrt{(-3)^2+10^2}}=\\frac{|-\\frac{3}{2}x_E+y_E|}{\\sqrt{\\left(-\\frac{3}{2}\\right)^2+1^2}}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On s&#8217;appuie ici sur le r\u00e9sultat qui dit que la distance d&#8217;un point de coordonn\u00e9e \\((\\alpha;\\beta)\\) \u00e0 la droite d&#8217;\u00e9quation cart\u00e9sienne \\(ax+by+c=0\\) est \u00e9gale \u00e0 \\( \\frac{|a\\alpha+b\\beta+c|}{\\sqrt{a^2+b^2}}\\), r\u00e9sultat vu en classe de Terminale (voir de 1\u00e8re en exercice pour certain\u00b7e\u00b7s).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&#8217;\u00e9galit\u00e9 pr\u00e9c\u00e9dente donne:$$\\sqrt{13}|-3x_E+10y_E|=2\\sqrt{109}\\left|-\\frac{3}{2}x_E+y_E\\right|$$c&#8217;est-\u00e0-dire:$$\\begin{array}{ll}-3x_E+10y_E &amp; = \\frac{2\\sqrt{109}}{\\sqrt{13}}\\left(-\\frac{3}{2}x_E+y_E\\right)\\\\\\text{ou}\\\\ -3x_E+10y_E &amp; = -\\frac{2\\sqrt{109}}{\\sqrt{13}}\\left(-\\frac{3}{2}x_E+y_E\\right)\\end{array}$$soit:$$x_E=\\frac{11-\\sqrt{1417}}{36}y_E \\text{ ou } x_E=\\frac{11+\\sqrt{1417}}{36}y_E.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Comme le rapport \\(x_E\/y_E\\) est positif, on retiendra:$$\\boxed{x_E=\\frac{11+\\sqrt{1417}}{36}y_E}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Bissectrice_de_AC_et_DC\"><\/span>Bissectrice de (AC) et (DC)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">La droite (DC) a pour \u00e9quation: \\(y=3\\). La distance de E \u00e0 (AD) doit \u00eatre \u00e9gale \u00e0 celle de E \u00e0 (AD):$$3-y_E=\\frac{|-3x_E+10y_E|}{\\sqrt{109}}.$$D&#8217;o\u00f9:$$\\boxed{x_E=\\frac{10+\\sqrt{109}}{3}y_E-\\sqrt{109}}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Des deux \u00e9galit\u00e9s encadr\u00e9es, on d\u00e9duit:<\/p>\n\n\n\n<p class=\"has-vivid-red-color has-text-color wp-block-paragraph\">$$y_E=\\frac{36\\sqrt{109}}{12\\sqrt{109}-\\sqrt{1417}+109}\\approx1,9.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D&#8217;o\u00f9:<\/p>\n\n\n\n<p class=\"has-vivid-red-color has-text-color wp-block-paragraph\">$$x_E=\\frac{\\sqrt{109}(11+\\sqrt{1417})}{12\\sqrt{109}-\\sqrt{1417}+109}\\approx2,6.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_longueur_EF\"><\/span>La longueur EF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Bon, l\u00e0, on va \u00eatre honn\u00eate, on n&#8217;a pas super envie de faire la m\u00eame chose pour le point F car&#8230; ben, c&#8217;est un peu long hein ? On va invoquer la d\u00e9esse de la sym\u00e9trie par rapport au milieu de la diagonale trac\u00e9e&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Les coordonn\u00e9es du milieu sont \\( I\\left( 5;\\frac{3}{2} \\right) \\) donc:$$\\begin{align}EF &amp; = 2IE \\\\ &amp; = 2\\sqrt{\\big(x_E-x_I \\big)^2 + \\big(y_E-y_I)^2}\\\\ &amp; = \\frac{\\sqrt{ 492728832\\sqrt{154453} &#8211; 23158255104\\sqrt{109} &#8211; 6035928192\\sqrt{1417} + 288677504448 }}{23544} \\\\ &amp;  \\approx 4,9\\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">J&#8217;avoue avoir utilis\u00e9<a href=\"https:\/\/www-fourier.ujf-grenoble.fr\/~parisse\/giac_fr.html\" target=\"_blank\" rel=\"noreferrer noopener\"> Xcas<\/a> pour la derni\u00e8re expression&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conclusion:<\/strong> la prochaine fois que je m&#8217;ennuie, je dois absolument arr\u00eater d&#8217;inventer des exercices aussi longs&#8230; \ud83d\ude42<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dans cet article, je fais suite \u00e0 celui publi\u00e9 hier, et le g\u00e9n\u00e9ralise dans le parall\u00e9logramme ci-dessous. Dans cette figure, les cercles trac\u00e9s sont tangents aux c\u00f4t\u00e9s du parall\u00e9logramme et \u00e0 la diagonale trac\u00e9e. Tout est parti du fait que je m&#8217;ennuyais en ce dimanche apr\u00e8s-midi&#8230;<\/p>\n","protected":false},"author":1,"featured_media":7837,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-7809","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Distance entre les centres de cercles dans un parall\u00e9logramme<\/title>\n<meta name=\"description\" content=\"Trouver la distance entre deux centres de cercles tangents \u00e0 une diagonale et aux c\u00f4t\u00e9s du parall\u00e9logramme dans lequel ils sont.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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