{"id":530,"date":"2018-09-01T11:11:23","date_gmt":"2018-09-01T09:11:23","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=530"},"modified":"2021-10-26T17:19:28","modified_gmt":"2021-10-26T15:19:28","slug":"calculer-la-longueur-dune-bissectrice-dans-un-triangle","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/calculer-la-longueur-dune-bissectrice-dans-un-triangle\/","title":{"rendered":"Longueur d&rsquo;une bissectrice dans un triangle"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Longueur d&rsquo;une bissectrice dans un triangle: consid\u00e9rons un triangle quelconque ABC ; posons alors AB = <em>c<\/em>, AC = <em>b&nbsp;<\/em>et BC = <em>a<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Posons \\(\\ell\\) la longueur de la bissectrice issue de C ; alors, on a:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"80\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01-300x80.png\" alt=\"Longueur d'une bissectrice dans un triangle\" class=\"wp-image-531\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01-300x80.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01-600x160.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01-768x205.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/longueur-bissectrice01.png 906w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">D\u00e9montrons ce r\u00e9sultat&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\">Longueur d&rsquo;une bissectrice dans un triangle: d\u00e9monstration (niveau : 1\u00e8re)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;aire du triangle:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>ABC est donn\u00e9e par la formule : \\[ \\mathcal{A}_{\\text{ABC}}=\\frac{1}{2}ab\\sin(2\\alpha). \\]<\/li><li>riangle ACI est donn\u00e9e par la formule : \\[ \\mathcal{A}_{\\text{ACI}}=\\frac{1}{2}b\\ell\\sin(\\alpha). \\]<\/li><li>BCI est donn\u00e9e par la formule : \\[ \\mathcal{A}_{\\text{BCI}}=\\frac{1}{2}a\\ell\\sin(\\alpha). \\]<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors : \\[&nbsp;\\begin{align*} &amp; \\mathcal{A}_{\\text{ABC}} = \\mathcal{A}_{\\text{ACI}}+\\mathcal{A}_{\\text{BCI}}\\\\&nbsp;\\Leftrightarrow\\ &amp; \\frac{1}{2}ab\\sin(2\\alpha) = \\frac{1}{2}b\\ell\\sin(\\alpha) + \\frac{1}{2}a\\ell\\sin(\\alpha)\\\\&nbsp;\\Leftrightarrow\\ &amp; ab\\sin(2\\alpha) =\\ell\\sin(\\alpha)(b + a)\\\\&nbsp;\\Leftrightarrow\\ &amp; \\ell = \\frac{ab}{a+b}\\times\\frac{\\sin(2\\alpha)}{\\sin\\alpha}. \\end{align*}\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or,&nbsp;\\[ \\sin(2\\alpha)=2\\sin\\alpha\\cos\\alpha.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;o\u00f9 :&nbsp;\\[\\ell = \\frac{ab}{a+b}\\times\\frac{2\\sin\\alpha\\cos\\alpha}{\\sin\\alpha},\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">et donc :&nbsp;\\[\\ell = \\frac{ab}{a+b}\\times 2\\cos\\alpha.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors que :&nbsp;\\[ \\ell^2 = \\frac{(ab)^2}{(a+b)^2}\\times 4\\cos^2\\alpha.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or,&nbsp;\\[ \\cos^2\\alpha = \\frac{1+\\cos(2\\alpha)}{2},\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d&rsquo;o\u00f9 : \\[ \\begin{align*} \\ell^2 &amp; = \\frac{(ab)^2}{(a+b)^2}\\times 4\\times\\frac{1+\\cos(2\\alpha)}{2}\\\\&nbsp;&amp; = \\frac{(ab)^2}{(a+b)^2}\\times 2(1+\\cos(2\\alpha))&nbsp;\\end{align*} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La relation d&rsquo;Al-Kashi dans le triangle ABC donne :&nbsp;\\[ c^2=a^2+b^2-2ab\\cos(2\\alpha).\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Donc :&nbsp;\\[ \\cos(2\\alpha)=\\frac{a^2+b^2-c^2}{2ab}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On a alors : \\[&nbsp;\\begin{align*}&nbsp;\\ell^2 &amp; = \\frac{(ab)^2}{(a+b)^2}\\times 2\\left(1+\\frac{a^2+b^2-c^2}{2ab}\\right)\\\\&nbsp;&amp; = \\frac{(ab)^2}{(a+b)^2}\\times 2\\left(\\frac{2ab+a^2+b^2-c^2}{2ab}\\right)\\\\&nbsp;&amp; = \\frac{ab}{(a+b)^2}(a^2+2ab+b^2-c^2)\\\\&nbsp;&amp; = \\frac{ab}{(a+b)^2}\\left[(a+b)^2-c^2\\right]\\\\&nbsp;&amp; = ab\\left(1-\\frac{c^2}{(a+b)^2}\\right)&nbsp;\\end{align*} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors (car \\(\\ell\\geq 0\\)):&nbsp;\\[ \\ell=\\sqrt{ab\\left(1-\\frac{c^2}{(a+b)^2}\\right)}.\\]<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Longueur-d_une-bissectrice-dans-un-triangle.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Longueur d&rsquo;une bissectrice dans un triangle<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Longueur-d_une-bissectrice-dans-un-triangle.pdf\" class=\"wp-block-file__button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Les sources \\(\\LaTeX\\) du document PDF<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Longueur-dune-bissectrice-dans-un-triangle.zip\">Longueur d&rsquo;une bissectrice dans un triangle<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Longueur-dune-bissectrice-dans-un-triangle.zip\" class=\"wp-block-file__button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Longueur d&rsquo;une bissectrice dans un triangle: consid\u00e9rons un triangle quelconque ABC ; posons alors AB = c, AC = b&nbsp;et BC = a. Posons \\(\\ell\\) la longueur de la bissectrice issue de C ; alors, on a: D\u00e9montrons ce r\u00e9sultat&#8230;<\/p>\n","protected":false},"author":1,"featured_media":3707,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[49,47,25,46],"class_list":["post-530","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-al-kashi","tag-bissectrice","tag-demonstration","tag-triangle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Longueur d&#039;une bissectrice dans un triangle - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Nous allons voir dans cet article comment calculer la longueur d&#039;une bissectrice dans un triangle: accessible d\u00e8s la 1\u00e8re sp\u00e9cialit\u00e9 maths.\" \/>\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\/2018\/09\/01\/calculer-la-longueur-dune-bissectrice-dans-un-triangle\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Longueur d&#039;une bissectrice dans un triangle - 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