{"id":16067,"date":"2023-02-17T05:33:42","date_gmt":"2023-02-17T04:33:42","guid":{"rendered":"https:\/\/college-willy-ronis.fr\/maths\/?p=16067"},"modified":"2023-03-06T10:41:24","modified_gmt":"2023-03-06T09:41:24","slug":"chapitre-11-le-theoreme-de-thales","status":"publish","type":"post","link":"https:\/\/college-willy-ronis.fr\/maths\/chapitre-11-le-theoreme-de-thales\/","title":{"rendered":"Chapitre 11 : Le th\u00e9or\u00e8me de Thal\u00e8s"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"16067\" class=\"elementor elementor-16067\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6edfc71f elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"6edfc71f\" data-element_type=\"section\" 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M65.4,11.1c-0.1,0.3,0.3,0.5,1.9-0.2s2.6-1.3,2.2-1.2s-0.9,0.4-2.5,0.8C65.3,10.9,65.5,10.8,65.4,11.1\tz M34.5,12.4c-0.2,0,2.1,0.8,3.3,0.9c1.2,0.1,2,0.1,2-0.2c0-0.3-0.1-0.5-1.6-0.4C36.6,12.8,34.7,12.4,34.5,12.4z M152.2,21.1\tc-0.1,0.1-2.4-0.3-7.5-0.3c-5,0-13.6-2.4-17.2-3.5c-3.6-1.1,10,3.9,16.5,4.1C150.5,21.6,152.3,21,152.2,21.1z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M269.6,18c-0.1-0.1-4.6,0.3-7.2,0c-7.3-0.7-17-3.2-16.6-2.9c0.4,0.3,13.7,3.1,17,3.3\tC267.7,18.8,269.7,18,269.6,18z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M227.4,9.8c-0.2-0.1-4.5-1-9.5-1.2c-5-0.2-12.7,0.6-12.3,0.5c0.3-0.1,5.9-1.8,13.3-1.2\tS227.6,9.9,227.4,9.8z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M204.5,13.4c-0.1-0.1,2-1,3.2-1.1c1.2-0.1,2,0,2,0.3c0,0.3-0.1,0.5-1.6,0.4\tC206.4,12.9,204.6,13.5,204.5,13.4z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M201,10.6c0-0.1-4.4,1.2-6.3,2.2c-1.9,0.9-6.2,3.1-6.1,3.1c0.1,0.1,4.2-1.6,6.3-2.6\tS201,10.7,201,10.6z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M154.5,26.7c-0.1-0.1-4.6,0.3-7.2,0c-7.3-0.7-17-3.2-16.6-2.9c0.4,0.3,13.7,3.1,17,3.3\tC152.6,27.5,154.6,26.8,154.5,26.7z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M41.9,19.3c0,0,1.2-0.3,2.9-0.1c1.7,0.2,5.8,0.9,8.2,0.7c4.2-0.4,7.4-2.7,7-2.6\tc-0.4,0-4.3,2.2-8.6,1.9c-1.8-0.1-5.1-0.5-6.7-0.4S41.9,19.3,41.9,19.3z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M75.5,12.6c0.2,0.1,2-0.8,4.3-1.1c2.3-0.2,2.1-0.3,2.1-0.5c0-0.1-1.8-0.4-3.4,0\tC76.9,11.5,75.3,12.5,75.5,12.6z\"\/>\n\t<path class=\"elementor-shape-fill\" d=\"M15.6,13.2c0-0.1,4.3,0,6.7,0.5c2.4,0.5,5,1.9,5,2c0,0.1-2.7-0.8-5.1-1.4\tC19.9,13.7,15.7,13.3,15.6,13.2z\"\/>\n<\/svg>\t\t<\/div>\n\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-772e019c\" data-id=\"772e019c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7c192745 elementor-widget elementor-widget-heading\" data-id=\"7c192745\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Chapitre 11<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-556d4114 elementor-widget elementor-widget-heading\" data-id=\"556d4114\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Le th\u00e9or\u00e8me de Thal\u00e8s<\/h1>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-485c7cb9 elementor-widget elementor-widget-text-editor\" data-id=\"485c7cb9\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-24190164\" data-id=\"24190164\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-76031458 elementor-widget elementor-widget-image\" data-id=\"76031458\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"256\" height=\"192\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/cours-maths-4eme.png\" class=\"attachment-medium size-medium wp-image-15214\" alt=\"Cours de math\u00e9matiques de 4\u00e8me\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-87db4dd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"87db4dd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e9f53d8\" data-id=\"e9f53d8\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-787a5ce elementor-widget elementor-widget-spacer\" data-id=\"787a5ce\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-20e7937 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"20e7937\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-11c2429\" data-id=\"11c2429\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-36d2fb8 elementor-widget elementor-widget-heading\" data-id=\"36d2fb8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Chapitre 11 : Le th\u00e9or\u00e8me de Thal\u00e8s<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-227cab5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"227cab5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-07313f3\" data-id=\"07313f3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-28c862c elementor-widget elementor-widget-spacer\" data-id=\"28c862c\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-828d39c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"828d39c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-91997a7\" data-id=\"91997a7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3c2a5d5 elementor-widget elementor-widget-heading\" data-id=\"3c2a5d5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">A) Le th\u00e9or\u00e8me de Thal\u00e8s<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d7635ae elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d7635ae\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f6c82bc\" data-id=\"f6c82bc\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-252c003 elementor-widget elementor-widget-text-editor\" data-id=\"252c003\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>Propri\u00e9t\u00e9 :<\/strong> $ABC$ est un triangle.<br \/>Si les points $M$ et $N$ sont des points respectifs des demi-droites $[AB)$ et $[AC)$ tels que les droites $(BC)$ et $(MN)$ sont parall\u00e8les, alors :<br \/>\\[\\dfrac{A\\textcolor{red}{M}}{A\\textcolor{blue}{B}}=\\dfrac{A\\textcolor{red}{N}}{A\\textcolor{blue}{C}}=\\dfrac{\\textcolor{red}{MN}}{\\textcolor{blue}{BC}}\\]\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-02f5e1c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"02f5e1c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-366906b\" data-id=\"366906b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bd1e77d elementor-widget elementor-widget-image\" data-id=\"bd1e77d\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"886\" height=\"341\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-1.png\" class=\"attachment-large size-large wp-image-16077\" alt=\"\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-1.png 886w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-1-300x115.png 300w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-1-768x296.png 768w\" sizes=\"auto, (max-width: 886px) 100vw, 886px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2c7eb40 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2c7eb40\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-44451eb\" data-id=\"44451eb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-893a256 elementor-widget elementor-widget-text-editor\" data-id=\"893a256\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>D\u00e9monstration :<\/strong> Les droites $(BC)$ et $(MN)$ \u00e9tant parall\u00e8les, les angles correspondants $\\widehat{ABC}$ et $\\widehat{AMN}$ et les angles correspondants $\\widehat{ACB}$ et $\\widehat{ANM}$ ont la m\u00eame mesure.<\/p><p>Ainsi, les triangles $ABC$ et $AMN$ ont leurs angles deux \u00e0 deux de m\u00eame mesure (on dit que les deux triangles sont semblables).<\/p><p>Donc le triangle $ABC$ est un agrandissement (ou une r\u00e9duction) du triangle $AMN$. On obtient donc les longueurs des c\u00f4t\u00e9s d&rsquo;un des triangles en multipliant par un nombre non nul les longueurs des c\u00f4t\u00e9s de l&rsquo;autre triangle. Ceci revient \u00e0 dire que le tableau ci-dessous est un tableau de proportionnalit\u00e9 :<\/p><p>On peut alors \u00e9crire les \u00e9galit\u00e9s suivantes :<br \/>\\[\\dfrac{AM}{AB}=\\dfrac{AN}{AC}=\\dfrac{MN}{BC}\\]\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2f416c9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2f416c9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0ceb842\" data-id=\"0ceb842\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a81b3fc elementor-widget elementor-widget-text-editor\" data-id=\"a81b3fc\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<strong>Remarques :<\/strong>\n<ul>\n \t<li>En pr\u00e9sence d&rsquo;une configuration de Thal\u00e8s, le tableau ci-dessous est un tableau de proportionnalit\u00e9 :<\/li>\n<\/ul>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16089 size-full\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-3.png\" alt=\"\" width=\"567\" height=\"78\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-3.png 567w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-3-300x41.png 300w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/>\n<ul>\n \t<li>Le triangle $AMN$ est un agrandissement ou une r\u00e9duction du triangle $ABC$.<\/span><\/li>\n \t<li>Le th\u00e9or\u00e8me de Thal\u00e8s permet de calculer des longueurs.<\/span><\/li>\n<\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a2b5dc0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a2b5dc0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-14f26f4\" data-id=\"14f26f4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8696317 elementor-widget elementor-widget-spacer\" data-id=\"8696317\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-31f0c13 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"31f0c13\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4a8c69f\" data-id=\"4a8c69f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ea307e4 elementor-widget elementor-widget-heading\" data-id=\"ea307e4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">B) Calculer une longueur avec le th\u00e9or\u00e8me de Thal\u00e8s<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-accba28 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"accba28\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d30f903\" data-id=\"d30f903\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cf7a6d8 elementor-widget elementor-widget-text-editor\" data-id=\"cf7a6d8\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>Exemple :<\/strong> Soit un triangle tel que $AC = 7$ cm, $AB = 2,5$ cm et $BC = 8$ cm.<br>Soit $M$ un point appartenant au segment $[AB]$ tel que $AM = 1,7$ cm et $N$ le point d&rsquo;intersection de la droite $(BC)$ avec la parall\u00e8le \u00e0 $(AC)$ passant par $M$.<\/p>\n<p>Calculer $BN$.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16080 size-full\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-4.png\" alt=\"\" width=\"355\" height=\"283\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-4.png 355w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-4-300x239.png 300w\" sizes=\"auto, (max-width: 355px) 100vw, 355px\" \/>$M\\in [AB]$<\/p>\n<p>$N\\in [BC]$<\/p>\n<p>Les droites $(MN)$ et $(AC)$ sont parall\u00e8les.<\/p>\n<p>D&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Thal\u00e8s :<\/p>\n<p>\\[\\dfrac{BM}{BA}=\\dfrac{BN}{BC}=\\dfrac{MN}{AC}\\]<br>\\[\\dfrac{0,8}{2,5}=\\dfrac{BN}{8}=\\dfrac{MN}{7}\\]\n<p>Donc :<br>\\[\\dfrac{0,8}{2,5}=\\dfrac{BN}{8}\\]\n<p>Ainsi :<br>\\[BN=\\dfrac{0,8\\times 8}{2,5}=2,56~\\text{cm}\\]\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d5a00b8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d5a00b8\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3740b98\" data-id=\"3740b98\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-09b33b8 elementor-widget elementor-widget-video\" data-id=\"09b33b8\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/_i6O0dkulzk&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-360e6c6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"360e6c6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-df4bfc1\" data-id=\"df4bfc1\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8654e89 elementor-widget elementor-widget-spacer\" data-id=\"8654e89\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3dbb415 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3dbb415\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-eb1302a\" data-id=\"eb1302a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-46dd0ad elementor-widget elementor-widget-heading\" data-id=\"46dd0ad\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">C) Justifier que deux droites ne sont pas parall\u00e8les<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-965a647 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"965a647\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-04bbbaf\" data-id=\"04bbbaf\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2f88a0e elementor-widget elementor-widget-text-editor\" data-id=\"2f88a0e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>Propri\u00e9t\u00e9 :<\/strong> $ABC$ est un triangle.<br \/>Si les points $M$ et $N$ sont des points respectifs des demi-droites $[AB)$ et $[AC)$ tels que $\\dfrac{A\\textcolor{red}{M}}{A\\textcolor{blue}{B}}\\neq \\dfrac{A\\textcolor{red}{N}}{A\\textcolor{blue}{C}} $ alors les droites $(BC)$ et $(MN)$ ne sont pas parall\u00e8les.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4d6aab1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4d6aab1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1dc730b\" data-id=\"1dc730b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0491840 elementor-widget elementor-widget-text-editor\" data-id=\"0491840\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>Exemple :<\/strong> On donne les longueurs suivantes :<br \/>$AB = 6,3$ cm ; $BC = 4,9$ cm ; $AE = 17$ cm et $DE = 7$ cm.<br \/>Les droites $(BD)$ et $(CE)$ sont-elles parall\u00e8les ? Justifier la r\u00e9ponse.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16081 size-full\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-5.png\" alt=\"\" width=\"434\" height=\"204\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-5.png 434w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2023\/02\/thales-emboite-5-300x141.png 300w\" sizes=\"auto, (max-width: 434px) 100vw, 434px\" \/><\/p><p>\\[<br \/>\\left.<br \/>\\begin{array}{ccccc}<br \/>\\dfrac{AB}{AC}&amp;=&amp;\\dfrac{6,3}{11,2}&amp;=&amp;\\dfrac{9}{16}\\\\<br \/>\\\\<br \/>\\dfrac{AD}{AE}&amp;=&amp;\\dfrac{10}{17}&amp;<br \/>\\end{array}<br \/>\\right\\}\\mbox{Donc}~~\\dfrac{AB}{AC}\\neq \\dfrac{AD}{AE}<br \/>\\]Donc les droites $(BD)$ et $(CE)$ ne sont pas parall\u00e8les.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1079dc1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1079dc1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-94b54ff\" data-id=\"94b54ff\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6d6c6d2 elementor-widget elementor-widget-video\" data-id=\"6d6c6d2\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/ycyZxZJWVb4&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5a5f606 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5a5f606\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9436db4\" data-id=\"9436db4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-808ac97 elementor-widget elementor-widget-spacer\" data-id=\"808ac97\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3b60d62 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3b60d62\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ba14760\" data-id=\"ba14760\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3e22c53 elementor-button-info elementor-align-center elementor-widget elementor-widget-button\" data-id=\"3e22c53\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-xl\" href=\"https:\/\/nuage02.apps.education.fr\/index.php\/s\/mibKrHPHKzjYt5g\" target=\"_blank\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-icon elementor-align-icon-left\">\n\t\t\t\t<i aria-hidden=\"true\" class=\"far fa-arrow-alt-circle-right\"><\/i>\t\t\t<\/span>\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Exercices corrig\u00e9s<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-af1f710 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"af1f710\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-82db080\" data-id=\"82db080\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4f122a4 elementor-widget elementor-widget-spacer\" data-id=\"4f122a4\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Chapitre 11 Le th\u00e9or\u00e8me de Thal\u00e8s Chapitre 11 : Le th\u00e9or\u00e8me de Thal\u00e8s A) Le th\u00e9or\u00e8me de Thal\u00e8s Propri\u00e9t\u00e9 : $ABC$ est un triangle.Si les points $M$ et $N$ sont des points respectifs des demi-droites $[AB)$ et $[AC)$ tels que les droites $(BC)$ et $(MN)$ sont parall\u00e8les, alors :[dfrac{Atextcolor{red}{M}}{Atextcolor{blue}{B}}=dfrac{Atextcolor{red}{N}}{Atextcolor{blue}{C}}=dfrac{textcolor{red}{MN}}{textcolor{blue}{BC}}] D\u00e9monstration : Les droites $(BC)$ et [&hellip;]<\/p>\n","protected":false},"author":7440,"featured_media":16071,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"default","site-content-layout":"page-builder","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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