{"id":58347,"date":"2020-07-07T11:15:15","date_gmt":"2020-07-07T10:15:15","guid":{"rendered":"https:\/\/groupe-reussite.fr\/?page_id=58347"},"modified":"2026-10-20T15:06:28","modified_gmt":"2026-10-20T14:06:28","slug":"cours-en-ligne-theoreme-de-thales","status":"publish","type":"page","link":"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-theoreme-de-thales\/","title":{"rendered":"Cours en ligne sur le th\u00e9or\u00e8me de Thales"},"content":{"rendered":"<div class=\"Latex_page\">\n<section class=\"has_ae_slider elementor-element elementor-section-full_width elementor-reverse-mobile elementor-section-stretched elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default elementor-section elementor-top-section\">\n<div class=\"elementor-container elementor-column-gap-no\">\n<div class=\"elementor-row\">\n<div class=\"elementor-column-wrap elementor-element-populated colone_elementor\">\n<p style=\"text-align: justify;\">\t\t<div data-elementor-type=\"page\" data-elementor-id=\"86176\" class=\"elementor elementor-86176\">\n\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-top-section elementor-element elementor-element-6f9fb2b3 elementor-section-content-bottom elementor-section-boxed elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"6f9fb2b3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5032621c sticky_badeau_colone strick_col ae-bg-gallery-type-default\" data-id=\"5032621c\" data-element_type=\"column\" data-e-type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-top-section elementor-element elementor-element-611bd75f elementor-section-boxed elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"611bd75f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-55718945 strick_col ae-bg-gallery-type-default\" data-id=\"55718945\" data-element_type=\"column\" data-e-type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5940f45c elementor-widget elementor-widget-heading\" data-id=\"5940f45c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-tage-mage\/\">Cours en ligne Tage Mage <\/a><\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f9bc725 elementor-widget elementor-widget-heading\" data-id=\"f9bc725\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\">Chapitres du sous-test 2 du Tage Mage<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-56c02cf0 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"56c02cf0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fractions\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fractions<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-390da5ff elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"390da5ff\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-pourcentages\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Pourcentages<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-da8d658 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"da8d658\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-algebre\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Alg\u00e8bre<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-490ac2aa elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"490ac2aa\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-figures-geometriques\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">G\u00e9om\u00e9trie<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3fd40310 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3fd40310\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-vitesse\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Vitesse<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6edf1c7e elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"6edf1c7e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-arithmetique\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Arithm\u00e9tique<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4f2c805 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"4f2c805\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-puissances\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Puissances<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-447b658 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"447b658\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-regle-de-trois\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Proportionnalit\u00e9<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-53863aa elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"53863aa\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-denombrement\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9nombrement<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-11741389 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"11741389\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-probabilites\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Probabilit\u00e9s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6ace8ace elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"6ace8ace\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-moyenne-arithmetique-moyenne-ponderee\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Moyenne<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-13f1db2f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"13f1db2f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-theoreme-de-pythagore\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Th\u00e9or\u00e8me de Pythagore<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-10d04684 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"10d04684\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-theoreme-de-thales\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Th\u00e9or\u00e8me de Thal\u00e8s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-691a3a8a elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"691a3a8a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-racines-carrees\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Racines Carr\u00e9es<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2a5f7bd6 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2a5f7bd6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/nous-contacter\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">CONTACTEZ-NOUS<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t\n<\/div>\n<div class=\"elementor-column-wrap elementor-element-populated colone_latex\" style=\"text-align: justify;\">\n<div class=\"style_pag\" style=\"color: #000; font-family: 'Lato', Sans-serif;\">\n<div class=\"elementor-text-editor elementor-clearfix\" style=\"color: #000000; font-family: Lato, sans-serif;\">\n<h1 class=\"Latex_page_title\" style=\"text-align: center;\">Le th\u00e9or\u00e8me de Thal\u00e8s : formule, r\u00e9ciproque et exemples<\/h1>\n<p><em>Par Seda Biber, le 20\/10\/2026<\/em><\/p>\n<\/div>\n<p dir=\"ltr\" style=\"text-align: justify;\">Le <strong>th\u00e9or\u00e8me de Thal\u00e8s<\/strong> permet de calculer une longueur quand deux droites parall\u00e8les coupent deux droites s\u00e9cantes. Les longueurs des deux triangles ainsi form\u00e9s sont alors proportionnelles. Prenons un triangle ABC, avec M sur la droite (AB) et N sur la droite (AC). Si (MN) est parall\u00e8le \u00e0 (BC), alors <strong>AM\/AB = AN\/AC = MN\/BC<\/strong>. Sa <strong>r\u00e9ciproque<\/strong> fait le chemin inverse : elle sert \u00e0 prouver que deux droites sont parall\u00e8les. On parle aussi de \u00ab propri\u00e9t\u00e9 de Thal\u00e8s \u00bb. En France, on l&rsquo;\u00e9tudie en 4e dans la configuration des triangles embo\u00eet\u00e9s, puis en 3e dans la configuration papillon. Avec le <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-theoreme-de-pythagore\/\">th\u00e9or\u00e8me de Pythagore<\/a>, c&rsquo;est l&rsquo;un des deux th\u00e9or\u00e8mes de base pour calculer des longueurs dans les <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-figures-geometriques\/\">figures g\u00e9om\u00e9triques<\/a>, au brevet comme au lyc\u00e9e. On le retrouve aussi au Tage Mage et au Score IAE, o\u00f9 il faut rep\u00e9rer vite les <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-conditions-minimales\/\">conditions minimales<\/a> pour conclure. Si les rapports vous posent probl\u00e8me, un professeur en <a href=\"https:\/\/groupe-reussite.fr\/cours-particuliers\/maths\/tous-niveaux\/france\/\">cours particuliers de maths<\/a> peut reprendre la m\u00e9thode avec vous.<\/p>\n<p class=\"colone_latex_ancres\" style=\"color: #000000; font-family: Lato, sans-serif; text-align: center;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-theoreme-de-thales\/\">R\u00e9sum\u00e9 de cours<\/a> <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-theoreme-de-thales\/\">Exercices et corrig\u00e9s<\/a><\/p>\n<p class=\"colone_latex_ancres\" style=\"color: #000000; font-family: Lato, sans-serif; text-align: center;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-tage-mage\/\">Cours en ligne du Tage Mage<\/a><\/p>\n<h2 dir=\"ltr\">R\u00e9sum\u00e9 : l&rsquo;essentiel sur le th\u00e9or\u00e8me de Thal\u00e8s<\/h2>\n<table class=\"table\" style=\"width: 100%; border-color: #0F2742; color: #0f2742; background: #ffffff;\">\n<tbody>\n<tr>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>\u00c9nonc\u00e9<\/strong><\/td>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>Ce que l&rsquo;on sait<\/strong><\/td>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>Ce que l&rsquo;on obtient<\/strong><\/td>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>\u00c0 quoi il sert<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\"><strong>Th\u00e9or\u00e8me de Thal\u00e8s<\/strong><\/td>\n<td style=\"border: 1px solid #0F2742;\">Les points sont align\u00e9s et deux droites sont parall\u00e8les<\/td>\n<td style=\"border: 1px solid #0F2742;\"><em>AM\/AB = AN\/AC = MN\/BC<\/em><\/td>\n<td style=\"border: 1px solid #0F2742;\">Calculer une longueur<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\"><strong>R\u00e9ciproque<\/strong><\/td>\n<td style=\"border: 1px solid #0F2742;\">Les points sont align\u00e9s dans le m\u00eame ordre et <em>AM\/AB = AN\/AC<\/em><\/td>\n<td style=\"border: 1px solid #0F2742;\">(MN) \/\/ (BC)<\/td>\n<td style=\"border: 1px solid #0F2742;\">Prouver que deux droites sont parall\u00e8les<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\"><strong>Contrapos\u00e9e<\/strong><\/td>\n<td style=\"border: 1px solid #0F2742;\">Les points sont align\u00e9s et <em>AM\/AB \u2260 AN\/AC<\/em><\/td>\n<td style=\"border: 1px solid #0F2742;\">(MN) et (BC) ne sont pas parall\u00e8les<\/td>\n<td style=\"border: 1px solid #0F2742;\">Prouver que deux droites ne sont pas parall\u00e8les<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"style_pag\" style=\"color: #000; font-family: 'Lato', Sans-serif;\">\n<p dir=\"ltr\">\t\t<div data-elementor-type=\"section\" data-elementor-id=\"186409\" class=\"elementor elementor-186409\">\n\t\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-inner-section elementor-element elementor-element-208557d4 elementor-section-content-bottom full_width elementor-section-full_width elementor-section-height-min-height elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"208557d4\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;gradient&quot;}\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-narrow\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-5c1307a9 ae-bg-gallery-type-default\" data-id=\"5c1307a9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a42a1b elementor-widget elementor-widget-heading\" data-id=\"a42a1b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">AVOIR LES MEILLEURS PROFS DE MATHS<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-53b6892b elementor-widget elementor-widget-heading\" data-id=\"53b6892b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">C'est gagner en autonomie<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-339aefc1 elementor-align-left elementor-mobile-align-center elementor-widget elementor-widget-button\" data-id=\"339aefc1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-md\" href=\"https:\/\/groupe-reussite.fr\/cours-particuliers\/maths\/tous-niveaux\/france\/\" target=\"_blank\" id=\"CTA_PRD_CP\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Cours de maths particulier<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-461d5d63 ae-bg-gallery-type-default\" data-id=\"461d5d63\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n<h2 dir=\"ltr\">Que dit le th\u00e9or\u00e8me de Thal\u00e8s ?<\/h2>\n<p dir=\"ltr\">Voici l&rsquo;\u00e9nonc\u00e9 complet. Soient deux droites (d) et (d&rsquo;) s\u00e9cantes en un point A. Les points B et M sont sur (d), les points C et N sont sur (d&rsquo;), et tous sont distincts de A.<\/p>\n<p dir=\"ltr\"><strong>Si les droites (MN) et (BC) sont parall\u00e8les, alors AM\/AB = AN\/AC = MN\/BC.<\/strong><\/p>\n<p dir=\"ltr\">Pour l&rsquo;appliquer, v\u00e9rifiez trois conditions :<\/p>\n<ul>\n<li>les points A, M, B sont align\u00e9s ;<\/li>\n<li>les points A, N, C sont align\u00e9s ;<\/li>\n<li>les droites (MN) et (BC) sont parall\u00e8les. C&rsquo;est en g\u00e9n\u00e9ral une donn\u00e9e de l&rsquo;\u00e9nonc\u00e9.<\/li>\n<\/ul>\n<p dir=\"ltr\">Le th\u00e9or\u00e8me fonctionne dans <strong>n&rsquo;importe quel triangle<\/strong>, qu&rsquo;il soit rectangle ou non. Seul le parall\u00e9lisme compte. Il existe deux configurations possibles.<\/p>\n<h3 dir=\"ltr\">La configuration triangle (4e)<\/h3>\n<p dir=\"ltr\">M est sur le segment [AB] et N sur le segment [AC]. Le petit triangle AMN est alors \u00ab embo\u00eet\u00e9 \u00bb dans le grand triangle ABC. Le point commun A est un sommet des deux triangles.<\/p>\n<figure id=\"attachment_58355\" aria-describedby=\"caption-attachment-58355\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"wp-image-58355 size-full\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/theoreme-thales-triangle.jpg\" alt=\"cours theoreme thales\" width=\"400\" height=\"209\" \/><figcaption id=\"caption-attachment-58355\" class=\"wp-caption-text\">Sur cette figure, le point commun s&rsquo;appelle O, M est sur [OA] et N sur [OB]. La formule s&rsquo;\u00e9crit donc OM\/OA = ON\/OB = MN\/AB.<\/figcaption><\/figure>\n<h3 dir=\"ltr\">La configuration papillon (3e)<\/h3>\n<p dir=\"ltr\">Le point A est <strong>entre<\/strong> M et B, et entre N et C. Les deux triangles AMN et ABC sont oppos\u00e9s par le sommet A et forment un papillon. L&rsquo;\u00e9galit\u00e9 des rapports est exactement la m\u00eame.<\/p>\n<figure id=\"attachment_58356\" aria-describedby=\"caption-attachment-58356\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-58356 size-full\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/figure-theoreme-thales.jpg\" alt=\"triangle invers\u00e9 th\u00e9or\u00e8me thales\" width=\"400\" height=\"249\" \/><figcaption id=\"caption-attachment-58356\" class=\"wp-caption-text\">Ici, O est entre M et A, et entre N et B. On a toujours OM\/OA = ON\/OB = MN\/AB.<\/figcaption><\/figure>\n<h2 dir=\"ltr\">Quelle est la formule du th\u00e9or\u00e8me de Thal\u00e8s ?<\/h2>\n<p dir=\"ltr\">La formule est <strong>AM\/AB = AN\/AC = MN\/BC<\/strong>. Chaque fraction compare un c\u00f4t\u00e9 du petit triangle AMN au c\u00f4t\u00e9 correspondant du grand triangle ABC. Pour calculer une longueur, on garde <strong>deux<\/strong> des trois rapports : celui qui contient l&rsquo;inconnue, et celui dont on conna\u00eet les deux longueurs. On termine par un produit en croix.<\/p>\n<p dir=\"ltr\">Par exemple, si AM = 3, AB = 5 et AC = 10, l&rsquo;\u00e9galit\u00e9 AM\/AB = AN\/AC donne 3\/5 = AN\/10. On en d\u00e9duit AN = 3 \u00d7 10 \/ 5 = 6.<\/p>\n<div style=\"background: #FFF8EC; border-left: 6px solid #C77F0C; padding: 15px; margin: 20px 0; color: #0f2742;\"><strong>&#x1f4a1; Astuce :<\/strong> avant d&rsquo;\u00e9crire la formule, notez les deux triangles l&rsquo;un au-dessus de l&rsquo;autre, en mettant les sommets correspondants dans le m\u00eame ordre : <em>AMN<\/em> en haut, <em>ABC<\/em> en bas. Les rapports se lisent alors colonne par colonne : AM\/AB, AN\/AC, MN\/BC. Mettez toujours le petit triangle au num\u00e9rateur, vous \u00e9viterez les rapports invers\u00e9s.<\/div>\n<h2 dir=\"ltr\">Comment r\u00e9diger le th\u00e9or\u00e8me de Thal\u00e8s au brevet ?<\/h2>\n<p dir=\"ltr\">Une copie doit citer les hypoth\u00e8ses avant d&rsquo;\u00e9crire la formule. Voici une phrase type que vous pouvez reprendre :<\/p>\n<blockquote>\n<p dir=\"ltr\">Les points A, M, B sont align\u00e9s, ainsi que les points A, N, C, et les droites (MN) et (BC) sont parall\u00e8les. D&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Thal\u00e8s, on a : AM\/AB = AN\/AC = MN\/BC.<\/p>\n<\/blockquote>\n<p dir=\"ltr\">Remplacez ensuite les longueurs connues par leurs valeurs, puis faites le calcul. N&rsquo;oubliez pas l&rsquo;unit\u00e9 dans la r\u00e9ponse finale.<\/p>\n<h2 dir=\"ltr\">Exemples : calculer une longueur avec Thal\u00e8s<\/h2>\n<h3 dir=\"ltr\">Exemple 1 : configuration triangle<\/h3>\n<p dir=\"ltr\">Dans le triangle LMK, R est sur [ML] et S est sur [MK], avec (RS) \/\/ (LK). On donne ML = 6 cm, MK = 8 cm, LK = 5 cm et MS = 6 cm. Calculez MR et RS.<\/p>\n<p dir=\"ltr\">Les points M, R, L sont align\u00e9s, ainsi que M, S, K, et (RS) \/\/ (LK). D&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Thal\u00e8s : MR\/ML = MS\/MK = RS\/LK.<\/p>\n<ul>\n<li>MR\/6 = 6\/8, donc MR = 6 \u00d7 6 \/ 8 = 36\/8 = <strong>4,5 cm<\/strong>.<\/li>\n<li>RS\/5 = 6\/8, donc RS = 5 \u00d7 6 \/ 8 = 30\/8 = <strong>3,75 cm<\/strong>.<\/li>\n<\/ul>\n<p dir=\"ltr\">Si vous avez du mal \u00e0 simplifier 36\/8, revoyez notre <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fractions\/\">fiche sur les fractions<\/a>.<\/p>\n<figure id=\"attachment_58358\" aria-describedby=\"caption-attachment-58358\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-58358 size-full\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/exemple-type-thales.jpg\" alt=\"theoreme thales exemple type\" width=\"400\" height=\"258\" \/><figcaption id=\"caption-attachment-58358\" class=\"wp-caption-text\">Exemple de calcul avec le th\u00e9or\u00e8me de Thal\u00e8s dans le triangle LMK, (RS) parall\u00e8le \u00e0 (LK)<\/figcaption><\/figure>\n<h3 dir=\"ltr\">Exemple 2 : configuration papillon<\/h3>\n<p dir=\"ltr\">Les droites (AC) et (BD) se coupent en E, et (AB) \/\/ (CD). On donne AB = 3 cm, BD = 9 cm et BE = 5 cm. Calculez CD.<\/p>\n<p dir=\"ltr\">Comme E est sur le segment [BD], ED = BD \u2212 BE = 9 \u2212 5 = 4 cm. Les points B, E, D sont align\u00e9s, ainsi que A, E, C, et (AB) \/\/ (CD). D&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Thal\u00e8s : ED\/EB = EC\/EA = CD\/AB.<\/p>\n<p dir=\"ltr\">Donc 4\/5 = CD\/3, et CD = 3 \u00d7 4 \/ 5 = <strong>2,4 cm<\/strong>.<\/p>\n<figure id=\"attachment_58360\" aria-describedby=\"caption-attachment-58360\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-58360 size-full\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/Triangle-inverse-thales.jpg\" alt=\"exemple type theoreme thales\" width=\"400\" height=\"249\" \/><figcaption id=\"caption-attachment-58360\" class=\"wp-caption-text\">Exemple de calcul avec le th\u00e9or\u00e8me de Thal\u00e8s en configuration papillon, (AB) parall\u00e8le \u00e0 (CD)<\/figcaption><\/figure>\n<h2 dir=\"ltr\">Que dit la r\u00e9ciproque du th\u00e9or\u00e8me de Thal\u00e8s ?<\/h2>\n<p dir=\"ltr\">La r\u00e9ciproque sert \u00e0 <strong>d\u00e9montrer que deux droites sont parall\u00e8les<\/strong>. Elle s&rsquo;\u00e9nonce ainsi :<\/p>\n<p dir=\"ltr\"><strong>Si les points A, M, B et A, N, C sont align\u00e9s dans le m\u00eame ordre, et si AM\/AB = AN\/AC, alors les droites (MN) et (BC) sont parall\u00e8les.<\/strong><\/p>\n<p style=\"color: #000000; font-family: Lato, sans-serif; text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-58362 alignleft\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/reciproque-thales-exemple.jpg\" alt=\"reciproque thales droite parralele \" width=\"321\" height=\"250\" \/><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-58364 aligncenter\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/reciproque-thales-triangle-inverse.jpg\" alt=\"exemple type reciproque thales\" width=\"400\" height=\"289\" \/><\/p>\n<p><em>R\u00e9ciproque du th\u00e9or\u00e8me de Thal\u00e8s : points A, M, B et A, N, C align\u00e9s dans le m\u00eame ordre, configuration triangle et configuration papillon<\/em><\/p>\n<p dir=\"ltr\">On compare seulement deux rapports, sans MN\/BC. Les deux fractions se calculent s\u00e9par\u00e9ment. On ne pose jamais l&rsquo;\u00e9galit\u00e9 d&rsquo;avance, puisque c&rsquo;est justement ce qu&rsquo;on cherche \u00e0 v\u00e9rifier.<\/p>\n<div style=\"background: #FFF8EC; border-left: 6px solid #C77F0C; padding: 15px; margin: 20px 0; color: #0f2742;\"><strong>&#x1f4a1; Astuce :<\/strong> la condition \u00ab dans le m\u00eame ordre \u00bb est indispensable. Si M est sur [AB] mais que N est de l&rsquo;autre c\u00f4t\u00e9 de A par rapport \u00e0 C, les rapports peuvent \u00eatre \u00e9gaux sans que les droites soient parall\u00e8les. Faites une figure \u00e0 main lev\u00e9e pour v\u00e9rifier la position des points.<\/div>\n<h3 dir=\"ltr\">Exemple 3 : prouver que deux droites sont parall\u00e8les<\/h3>\n<p dir=\"ltr\" style=\"text-align: justify;\">Les points A, B, C sont align\u00e9s dans cet ordre, ainsi que A, D, E. On donne AB = 8 cm, BC = 4 cm, AD = 10 cm et DE = 5 cm. Les droites (BD) et (CE) sont-elles parall\u00e8les ?<\/p>\n<p dir=\"ltr\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-58366 aligncenter\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/exemple-reciproque-thales.jpg\" alt=\"triangle theoreme thales\" width=\"400\" height=\"240\" \/><\/p>\n<p>On calcule d&rsquo;abord AC = AB + BC = 8 + 4 = 12 cm et AE = AD + DE = 10 + 5 = 15 cm.<\/p>\n<p dir=\"ltr\">Les rapports sont \u00e9gaux et les points sont align\u00e9s dans le m\u00eame ordre. D&rsquo;apr\u00e8s la r\u00e9ciproque du th\u00e9or\u00e8me de Thal\u00e8s, <strong>(BD) \/\/ (CE)<\/strong>.<\/p>\n<h3 dir=\"ltr\">Exemple 4 : prouver que deux droites ne sont pas parall\u00e8les<\/h3>\n<p dir=\"ltr\">Les droites (GL) et (FP) se coupent en T. On donne TG = 1,8 cm, TL = 6 cm, TF = 3 cm et TP = 9 cm. Les droites (FG) et (LP) sont-elles parall\u00e8les ?<\/p>\n<p dir=\"ltr\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-58367 aligncenter\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/uploads\/2020\/07\/dernier-exemple-reciproque-thales.jpg\" alt=\"reciproque theoreme thales\" width=\"400\" height=\"202\" \/><\/p>\n<ul>\n<li>TG\/TL = 1,8\/6 = 3\/10 = 9\/30<\/li>\n<li>TF\/TP = 3\/9 = 1\/3 = 10\/30<\/li>\n<\/ul>\n<p dir=\"ltr\">Les rapports sont diff\u00e9rents. Si les droites \u00e9taient parall\u00e8les, le th\u00e9or\u00e8me de Thal\u00e8s imposerait des rapports \u00e9gaux. Donc <strong>(FG) et (LP) ne sont pas parall\u00e8les<\/strong>. Ce raisonnement s&rsquo;appelle la contrapos\u00e9e du th\u00e9or\u00e8me de Thal\u00e8s.<\/p>\n<p dir=\"ltr\">Pour vous entra\u00eener sur d&rsquo;autres figures, consultez nos <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-theoreme-de-thales\/\">exercices corrig\u00e9s sur le th\u00e9or\u00e8me de Thal\u00e8s et sa r\u00e9ciproque<\/a>.<\/p>\n<h2 dir=\"ltr\">Quelles erreurs \u00e9viter avec le th\u00e9or\u00e8me de Thal\u00e8s ?<\/h2>\n<table class=\"table\" style=\"width: 100%; border-color: #0F2742; color: #0f2742; background: #ffffff;\">\n<tbody>\n<tr>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>Erreur fr\u00e9quente<\/strong><\/td>\n<td style=\"background: #0F2742; color: #ffffff; border: 1px solid #0F2742; text-align: left;\"><strong>Ce qu&rsquo;il faut faire<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">Confondre Thal\u00e8s et Pythagore<\/td>\n<td style=\"border: 1px solid #0F2742;\">Thal\u00e8s demande des droites parall\u00e8les et utilise des rapports. Pythagore demande un triangle rectangle et utilise des carr\u00e9s.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">M\u00e9langer les c\u00f4t\u00e9s dans les rapports<\/td>\n<td style=\"border: 1px solid #0F2742;\">\u00c9crire les deux triangles l&rsquo;un sous l&rsquo;autre avec les sommets correspondants dans le m\u00eame ordre.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">\u00c9crire MN\/BC = AM\/MB<\/td>\n<td style=\"border: 1px solid #0F2742;\">Toujours partir du point commun A : AM\/AB, et jamais AM\/MB.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">Utiliser la r\u00e9ciproque pour calculer une longueur<\/td>\n<td style=\"border: 1px solid #0F2742;\">La r\u00e9ciproque ne sert qu&rsquo;\u00e0 prouver un parall\u00e9lisme. Pour calculer une longueur, on utilise le th\u00e9or\u00e8me.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">Oublier de citer les hypoth\u00e8ses<\/td>\n<td style=\"border: 1px solid #0F2742;\">Nommer les points align\u00e9s et les droites parall\u00e8les avant d&rsquo;\u00e9crire la formule.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #0F2742;\">Comparer des rapports en \u00e9criture d\u00e9cimale arrondie<\/td>\n<td style=\"border: 1px solid #0F2742;\">Simplifier les fractions ou les mettre au m\u00eame d\u00e9nominateur, comme dans l&rsquo;exemple 4.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\">\t\t<div data-elementor-type=\"section\" data-elementor-id=\"205992\" class=\"elementor elementor-205992\">\n\t\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-inner-section elementor-element elementor-element-df11d1e elementor-section-content-bottom full_width elementor-section-full_width elementor-section-height-min-height elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"df11d1e\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;gradient&quot;}\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-narrow\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-54ab74af ae-bg-gallery-type-default\" data-id=\"54ab74af\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c1ca7d elementor-widget elementor-widget-heading\" data-id=\"2c1ca7d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">STAGE DE MATHS PENDANT LES VACANCES<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3e600302 elementor-widget elementor-widget-heading\" data-id=\"3e600302\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Gagner en s\u00e9r\u00e9nit\u00e9 et progresser en confiance<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-598c6a68 elementor-align-left elementor-mobile-align-center elementor-widget elementor-widget-button\" data-id=\"598c6a68\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-md\" href=\"https:\/\/groupe-reussite.fr\/stages\/maths\/tous-niveaux\/france\/toutes-periodes\/\" target=\"_blank\" id=\"CTA_PRD_STAGE\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Stage maths vacances<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-14415dfd ae-bg-gallery-type-default\" data-id=\"14415dfd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n<h2 dir=\"ltr\">FAQ : questions fr\u00e9quentes sur le th\u00e9or\u00e8me de Thal\u00e8s<\/h2>\n        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Qu&#039;est-ce que le th\u00e9or\u00e8me de Thal\u00e8s en une phrase ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tSi deux droites parall\u00e8les coupent deux droites s\u00e9cantes en A, les longueurs des deux triangles form\u00e9s sont proportionnelles : AM\/AB = AN\/AC = MN\/BC.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Quelle diff\u00e9rence entre la propri\u00e9t\u00e9 de Thal\u00e8s et le th\u00e9or\u00e8me de Thal\u00e8s ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tAucune : les deux expressions d\u00e9signent le m\u00eame \u00e9nonc\u00e9. On parle de r\u00e9ciproque de la propri\u00e9t\u00e9 de Thal\u00e8s comme de r\u00e9ciproque du th\u00e9or\u00e8me.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Quand utiliser Thal\u00e8s plut\u00f4t que Pythagore ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tUtilisez Thal\u00e8s quand la figure contient deux droites parall\u00e8les et deux droites s\u00e9cantes. Utilisez Pythagore quand vous travaillez dans un triangle rectangle et que vous connaissez deux de ses c\u00f4t\u00e9s.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Le th\u00e9or\u00e8me de Thal\u00e8s fonctionne-t-il dans un triangle rectangle ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tOui. Le th\u00e9or\u00e8me s'applique dans tout triangle, rectangle ou non. La seule condition est le parall\u00e9lisme des droites (MN) et (BC).                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>En quelle classe \u00e9tudie-t-on le th\u00e9or\u00e8me de Thal\u00e8s ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tDans le programme de cycle 4 actuellement en vigueur en France, le th\u00e9or\u00e8me et sa r\u00e9ciproque sont vus en 4e dans la configuration des triangles embo\u00eet\u00e9s, puis en 3e dans la configuration papillon. Un nouveau programme de cycle 4 entrera en vigueur en 4e \u00e0 la rentr\u00e9e 2027 et en 3e \u00e0 la rentr\u00e9e 2028.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Le Thales&#039;s theorem anglais est-il le m\u00eame th\u00e9or\u00e8me ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tPas toujours. En anglais, <em>Thales's theorem<\/em> d\u00e9signe souvent un autre r\u00e9sultat : un triangle inscrit dans un cercle, dont un c\u00f4t\u00e9 est un diam\u00e8tre, est rectangle. Le th\u00e9or\u00e8me de Thal\u00e8s fran\u00e7ais correspond plut\u00f4t \u00e0 l'<em>intercept theorem<\/em>.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t\n<script type=\"application\/ld+json\">\n    {\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Qu'est-ce que le th\u00e9or\u00e8me de Thal\u00e8s en une phrase ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Si deux droites parall\u00e8les coupent deux droites s\u00e9cantes en A, les longueurs des deux triangles form\u00e9s sont proportionnelles : AM\/AB = AN\/AC = MN\/BC.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Quelle diff\u00e9rence entre la propri\u00e9t\u00e9 de Thal\u00e8s et le th\u00e9or\u00e8me de Thal\u00e8s ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Aucune : les deux expressions d\u00e9signent le m\u00eame \u00e9nonc\u00e9. On parle de r\u00e9ciproque de la propri\u00e9t\u00e9 de Thal\u00e8s comme de r\u00e9ciproque du th\u00e9or\u00e8me.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Quand utiliser Thal\u00e8s plut\u00f4t que Pythagore ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Utilisez Thal\u00e8s quand la figure contient deux droites parall\u00e8les et deux droites s\u00e9cantes. Utilisez Pythagore quand vous travaillez dans un triangle rectangle et que vous connaissez deux de ses c\u00f4t\u00e9s.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Le th\u00e9or\u00e8me de Thal\u00e8s fonctionne-t-il dans un triangle rectangle ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Oui. Le th\u00e9or\u00e8me s'applique dans tout triangle, rectangle ou non. La seule condition est le parall\u00e9lisme des droites (MN) et (BC).\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"En quelle classe \u00e9tudie-t-on le th\u00e9or\u00e8me de Thal\u00e8s ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Dans le programme de cycle 4 actuellement en vigueur en France, le th\u00e9or\u00e8me et sa r\u00e9ciproque sont vus en 4e dans la configuration des triangles embo\u00eet\u00e9s, puis en 3e dans la configuration papillon. Un nouveau programme de cycle 4 entrera en vigueur en 4e \u00e0 la rentr\u00e9e 2027 et en 3e \u00e0 la rentr\u00e9e 2028.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Le Thales's theorem anglais est-il le m\u00eame th\u00e9or\u00e8me ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Pas toujours. En anglais, &lt;em&gt;Thales's theorem&lt;\/em&gt; d\u00e9signe souvent un autre r\u00e9sultat : un triangle inscrit dans un cercle, dont un c\u00f4t\u00e9 est un diam\u00e8tre, est rectangle. Le th\u00e9or\u00e8me de Thal\u00e8s fran\u00e7ais correspond plut\u00f4t \u00e0 l'&lt;em&gt;intercept theorem&lt;\/em&gt;.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t    ]\n}\n<\/script>\n\n<\/div>\n<div class=\"style_pag\" style=\"color: #000; font-family: 'Lato', Sans-serif;\">\n<h2 dir=\"ltr\">Sources<\/h2>\n<ul dir=\"ltr\">\n<li>\u00c9duscol, <a href=\"https:\/\/eduscol.education.fr\/sites\/default\/files\/2020-10\/fiche-college-3e-1280085-pdf-1655.pdf\"><em>Attendus de fin d\u2019ann\u00e9e, math\u00e9matiques, 3e<\/em><\/a> : \u00ab le th\u00e9or\u00e8me de Thal\u00e8s et sa r\u00e9ciproque dans la configuration papillon \u00bb.<\/li>\n<li>Acad\u00e9mie de Bordeaux, <a href=\"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-content\/uploads\/sites\/3\/2019\/11\/Espace-et-G%C3%A9om%C3%A9trie-en-Quatri%C3%A8me.pdf\"><em>Espace et g\u00e9om\u00e9trie en quatri\u00e8me<\/em><\/a> : \u00ab le th\u00e9or\u00e8me de Thal\u00e8s et sa r\u00e9ciproque (configurations des triangles embo\u00eet\u00e9s) \u00bb.<\/li>\n<li>Acad\u00e9mie de Bordeaux, <a href=\"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/publication-du-nouveau-programme-cycle-4\/\"><em>Publication du nouveau programme cycle 4<\/em><\/a> (BO du 5 mars 2026) : calendrier d\u2019entr\u00e9e en vigueur par niveau.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Ne rel\u00e2chez pas vos efforts et assurez-vous de conna\u00eetre sur le bout des doigts toutes les notions de math\u00e9matiques pr\u00e9sentes dans le sous-test 2 Tage Mage, en utilisant nos cours en ligne du Tage Mage, dont en voici quelques-uns :<\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"text-align: justify;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-racines-carrees\/\">Cours sur les racines carr\u00e9es<\/a><\/li>\n<li style=\"text-align: justify;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-pourcentages\/\">R\u00e9sum\u00e9 de cours sur les pourcentages<\/a><\/li>\n<li style=\"text-align: justify;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-algebre\/\">Comprendre l&rsquo;alg\u00e8bre<\/a><\/li>\n<li style=\"text-align: justify;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-racines-carrees\/\">Exercices sur les racines carr\u00e9es<\/a><\/li>\n<li style=\"text-align: justify;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-fractions\/\">Entra\u00eenement sur les fractions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Le th\u00e9or\u00e8me de Thal\u00e8s : formule, r\u00e9ciproque et exemples Par Seda Biber, le 20\/10\/2026 Le th\u00e9or\u00e8me de Thal\u00e8s permet de calculer une longueur quand deux droites parall\u00e8les coupent deux droites s\u00e9cantes. Les longueurs des deux triangles ainsi form\u00e9s sont alors proportionnelles. Prenons un triangle ABC, avec M sur la droite (AB) et N sur la [&hellip;]<\/p>\n","protected":false},"author":6,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ocean_post_layout":"full-screen","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"ressource","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"on","ocean_disable_heading":"on","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"on","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","footnotes":""},"class_list":["post-58347","page","type-page","status-publish","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - 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