{"id":18001,"date":"2024-10-05T06:01:14","date_gmt":"2024-10-05T04:01:14","guid":{"rendered":"https:\/\/college-willy-ronis.fr\/maths\/?p=18001"},"modified":"2025-01-14T07:21:51","modified_gmt":"2025-01-14T06:21:51","slug":"caracteristique-deuler-dun-polyedre","status":"publish","type":"post","link":"https:\/\/college-willy-ronis.fr\/maths\/caracteristique-deuler-dun-polyedre\/","title":{"rendered":"Caract\u00e9ristique d&rsquo;Euler d&rsquo;un poly\u00e8dre"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"18001\" class=\"elementor elementor-18001\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-550508b1 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"550508b1\" data-element_type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;gradient&quot;,&quot;shape_divider_bottom&quot;:&quot;wave-brush&quot;}\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t<div class=\"elementor-shape elementor-shape-bottom\" data-negative=\"false\">\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 283.5 27.8\" preserveAspectRatio=\"none\">\n\t<path class=\"elementor-shape-fill\" d=\"M283.5,9.7c0,0-7.3,4.3-14,4.6c-6.8,0.3-12.6,0-20.9-1.5c-11.3-2-33.1-10.1-44.7-5.7\ts-12.1,4.6-18,7.4c-6.6,3.2-20,9.6-36.6,9.3C131.6,23.5,99.5,7.2,86.3,8c-1.4,0.1-6.6,0.8-10.5,2c-3.8,1.2-9.4,3.8-17,4.7\tc-3.2,0.4-8.3,1.1-14.2,0.9c-1.5-0.1-6.3-0.4-12-1.6c-5.7-1.2-11-3.1-15.8-3.7C6.5,9.2,0,10.8,0,10.8V0h283.5V9.7z M260.8,11.3\tc-0.7-1-2-0.4-4.3-0.4c-2.3,0-6.1-1.2-5.8-1.1c0.3,0.1,3.1,1.5,6,1.9C259.7,12.2,261.4,12.3,260.8,11.3z 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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-792ecd65\" data-id=\"792ecd65\" 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-46259b32 elementor-widget elementor-widget-heading\" data-id=\"46259b32\" 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\">Section d'excellence<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3cae1c95 elementor-widget elementor-widget-heading\" data-id=\"3cae1c95\" 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\">Caract\u00e9ristique d'Euler<\/h1>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4df5ff31 elementor-widget elementor-widget-text-editor\" data-id=\"4df5ff31\" 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-745e295a\" data-id=\"745e295a\" 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-1d1035c8 elementor-widget elementor-widget-image\" data-id=\"1d1035c8\" 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-7f43bc2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7f43bc2\" 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-975cbbe\" data-id=\"975cbbe\" 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-6c82120 elementor-widget elementor-widget-spacer\" data-id=\"6c82120\" 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-77f1d8c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"77f1d8c\" 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-76e95f4\" data-id=\"76e95f4\" 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-81d868c elementor-widget elementor-widget-heading\" data-id=\"81d868c\" 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\">A) Poly\u00e8dres convexes<\/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-456bc4d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"456bc4d\" 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-baa01ad\" data-id=\"baa01ad\" 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-c488a15 elementor-widget elementor-widget-text-editor\" data-id=\"c488a15\" 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\u00e9finition :<\/strong> Un<strong> poly\u00e8dre<\/strong> est un solide dont toutes les faces sont des polygones.<\/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-87a73db elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"87a73db\" 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-d44f41d\" data-id=\"d44f41d\" 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-3086e38 elementor-widget elementor-widget-text-editor\" data-id=\"3086e38\" 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\u00e9finition :<\/strong> Un poly\u00e8dre est <strong>convexe<\/strong> si toutes ses diagonales sont enti\u00e8rement contenues dans son int\u00e9rieur.<\/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-61e0a42 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"61e0a42\" 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-50 elementor-top-column elementor-element elementor-element-6c91079\" data-id=\"6c91079\" 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-0a3030e elementor-widget elementor-widget-text-editor\" data-id=\"0a3030e\" 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 :\u00a0<\/strong>Un poly\u00e8dre convexe (Dod\u00e9ca\u00e8dre r\u00e9gulier).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-18033 aligncenter\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/dodecaedre-regulier.png\" alt=\"un dod\u00e9ca\u00e8dre r\u00e9gulier\" width=\"196\" height=\"187\" \/><\/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<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-837d504\" data-id=\"837d504\" 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-45207bd elementor-widget elementor-widget-text-editor\" data-id=\"45207bd\" 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>Un poly\u00e8dre non convexe (Icosa\u00e8dre de Jessen).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-18034 aligncenter\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/icosaedre-Jessen.png\" alt=\"Icosa\u00e8dre de Jessen\" width=\"214\" height=\"202\" \/><\/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-96098de elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"96098de\" 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-a6c87b4\" data-id=\"a6c87b4\" 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-d733183 elementor-widget elementor-widget-spacer\" data-id=\"d733183\" 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-bbe16ef elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bbe16ef\" 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-4a6b296\" data-id=\"4a6b296\" 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-f77b004 elementor-widget elementor-widget-heading\" data-id=\"f77b004\" 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\">B) Caract\u00e9ristique d'Euler d'un poly\u00e8dre convexe<\/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-8682ae7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8682ae7\" 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-bf96998\" data-id=\"bf96998\" 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-6d75ee8 elementor-widget elementor-widget-text-editor\" data-id=\"6d75ee8\" 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>Les \u00e9l\u00e8ves se sont alors int\u00e9ress\u00e9s \u00e0 la caract\u00e9ristique d&rsquo;Euler d&rsquo;un poly\u00e8dre convexe.<\/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-e3e4b0e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e3e4b0e\" 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-70874a8\" data-id=\"70874a8\" 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-8bbc69d elementor-widget elementor-widget-text-editor\" data-id=\"8bbc69d\" 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\u00e9finition :<\/strong> La <strong>caract\u00e9ristique d&rsquo;Euler<\/strong> est la quantit\u00e9 $s-a+f$ o\u00f9 $s$ est le nombre de sommets, $a$ le nombre d&rsquo;ar\u00eates et $f$ le nombre de faces.<\/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-843cfd0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"843cfd0\" 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-e732adc\" data-id=\"e732adc\" 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-1fd5f00 elementor-widget elementor-widget-text-editor\" data-id=\"1fd5f00\" 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>Pour les aider \u00e0 \u00e9tablir une conjecture, ils disposaient de poly\u00e8dres convexes pos\u00e9s sur un \u00eelot de la salle mais \u00e9galement de G\u00e9oG\u00e9bra.<\/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-cc12cf2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cc12cf2\" 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-07de89f\" data-id=\"07de89f\" 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-643b9d8 elementor-widget elementor-widget-text-editor\" data-id=\"643b9d8\" 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>Exemples :<\/strong><\/p><p style=\"text-align: center;\">Cube adouci<\/p><p><iframe loading=\"lazy\" style=\"border: 0px;\" title=\"cube adouci\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/mqfpc8wv\/width\/881\/height\/508\/border\/888888\/sfsb\/true\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" width=\"881px\" height=\"508px\" scrolling=\"no\"> <\/iframe><\/p><p style=\"text-align: center;\">Dod\u00e9ca\u00e8dre rhombique<br \/><iframe loading=\"lazy\" style=\"border: 1px solid #e4e4e4; border-radius: 4px;\" src=\"https:\/\/www.geogebra.org\/classic\/n8r5akax?embed\" width=\"800\" height=\"600\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-9593a1e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9593a1e\" 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-85903ad\" data-id=\"85903ad\" 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-b193485 elementor-widget elementor-widget-text-editor\" data-id=\"b193485\" 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>Les \u00e9l\u00e8ves finissent par conjecturer que la caract\u00e9ristique d&rsquo;Euler pour les poly\u00e8dres convexes est toujours \u00e9gale \u00e0 2.<\/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-28d4931 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"28d4931\" 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-4a9f109\" data-id=\"4a9f109\" 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-6778d59 elementor-widget elementor-widget-text-editor\" data-id=\"6778d59\" 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>Th\u00e9or\u00e8me de Descartes-Euler :<\/strong> La quantit\u00e9 $s-a+f$ vaut 2 pour un poly\u00e8dre convexe.<\/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-db514f5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"db514f5\" 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-aa9411a\" data-id=\"aa9411a\" 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-8f6383c elementor-widget elementor-widget-text-editor\" data-id=\"8f6383c\" 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>Le th\u00e9or\u00e8me est formul\u00e9 par Leonhard Euler en 1752. Il semble cependant que Descartes ait prouv\u00e9 une relation<br \/>analogue dans un trait\u00e9 jamais publi\u00e9. C\u2019est la raison pour laquelle cette relation porte ce double nom.<\/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-a409b05 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a409b05\" 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-d1a65b1\" data-id=\"d1a65b1\" 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-02d27ff elementor-widget elementor-widget-text-editor\" data-id=\"02d27ff\" 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>Remarque :<\/strong> ce th\u00e9or\u00e8me est vrai uniquement pour les <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Genre_(math%C3%A9matiques)\" target=\"_blank\" rel=\"noopener\">poly\u00e8dres convexes de genre 0<\/a>.<\/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-f69b700 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f69b700\" 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-d0e1c1e\" data-id=\"d0e1c1e\" 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-a7c9aeb elementor-widget elementor-widget-spacer\" data-id=\"a7c9aeb\" 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-82023e4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"82023e4\" 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-bacf5fe\" data-id=\"bacf5fe\" 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-2174fc4 elementor-widget elementor-widget-heading\" data-id=\"2174fc4\" 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\">C) D\u00e9monstration de la conjecture<\/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-a0d69cb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a0d69cb\" 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-ddc5595\" data-id=\"ddc5595\" 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-2e21c0b elementor-widget elementor-widget-text-editor\" data-id=\"2e21c0b\" 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>Pour d\u00e9montrer cette conjecture, il a fallu d\u00e9former le poly\u00e8dre, en l\u2019aplatissant et en \u00e9cartant<br \/>vers l\u2019ext\u00e9rieur les c\u00f4t\u00e9s de cette face manquante. Nous obtenons alors le graphe planaire du poly\u00e8dre. En consid\u00e9rant<br \/>que tout l\u2019ext\u00e9rieur du graphe obtenu repr\u00e9sente la face enlev\u00e9e au poly\u00e8dre de d\u00e9part, le nombre de sommets,<br \/>d\u2019ar\u00eates et de faces n\u2019a pas chang\u00e9. Il suffit donc de d\u00e9montrer que la caract\u00e9ristique d\u2019Euler d\u2019un graphe planaire est<br \/>toujours \u00e9gale \u00e0 2.<\/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-68636ed elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68636ed\" 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-0e35d69\" data-id=\"0e35d69\" 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-0397ea4 elementor-widget elementor-widget-text-editor\" data-id=\"0397ea4\" 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> graphe planaire d&rsquo;un cube.<\/p><p>\u00a0<\/p><p><iframe loading=\"lazy\" style=\"border: 1px solid #e4e4e4; border-radius: 4px;\" src=\"https:\/\/www.geogebra.org\/classic\/ugxffbsz?embed\" width=\"800\" height=\"600\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-def7902 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"def7902\" 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-c6d460f\" data-id=\"c6d460f\" 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-bd9ed46 elementor-widget elementor-widget-text-editor\" data-id=\"bd9ed46\" 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> graphes planaires de certains poly\u00e8dres.<\/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-790f014 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"790f014\" 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-ed8fc41\" data-id=\"ed8fc41\" 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-dc5eec6 elementor-widget elementor-widget-image\" data-id=\"dc5eec6\" 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=\"587\" height=\"197\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphes-planaires.png\" class=\"attachment-large size-large wp-image-18032\" alt=\"graphes planaires de certains poly\u00e8dres\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphes-planaires.png 587w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphes-planaires-300x101.png 300w\" sizes=\"auto, (max-width: 587px) 100vw, 587px\" \/>\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-f9f7392 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f9f7392\" 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-096d99a\" data-id=\"096d99a\" 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-3368b3c elementor-widget elementor-widget-spacer\" data-id=\"3368b3c\" 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-ee94222 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ee94222\" 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-5156a6d\" data-id=\"5156a6d\" 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-d149b98 elementor-widget elementor-widget-text-editor\" data-id=\"d149b98\" 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\u00e9finition :<\/strong> Un <strong>graphe planaire<\/strong> est obtenu de la fa\u00e7on suivante : on choisit des points du plan appel\u00e9s sommets. On\u00a0peut ensuite choisir de relier les points distincts par des segments appel\u00e9s ar\u00eates, telles qu\u2019elles ne s\u2019intersectent pas.<\/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-3dc455e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3dc455e\" 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-a64366c\" data-id=\"a64366c\" 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-ad26374 elementor-widget elementor-widget-text-editor\" data-id=\"ad26374\" 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\u00e9finition :<\/strong> On appelle face du graphe une r\u00e9gion du plan entour\u00e9e par des segments.<\/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-a04f004 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a04f004\" 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-b4d14e7\" data-id=\"b4d14e7\" 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-ddbc1eb elementor-widget elementor-widget-text-editor\" data-id=\"ddbc1eb\" 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>Remarque :<\/strong> Lors du d\u00e9nombrement des faces, il ne faut pas oublier la face ext\u00e9rieure (celle qui est infinie).<\/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-353c04c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"353c04c\" 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-1e0df14\" data-id=\"1e0df14\" 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-3f3ad9b elementor-widget elementor-widget-text-editor\" data-id=\"3f3ad9b\" 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> Un graphe planaire.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-18041 size-full\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphe-planaire.png\" alt=\"Un graphe planaire\" width=\"469\" height=\"268\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphe-planaire.png 469w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/graphe-planaire-300x171.png 300w\" sizes=\"auto, (max-width: 469px) 100vw, 469px\" \/><\/p><p>Ce graphe poss\u00e8de 7 sommets, 8 ar\u00eates et 3 faces (2 faces + la face infinie).<\/p><p>Sa caract\u00e9ristique d&rsquo;Euler, not\u00e9 $\\chi$ vaut alors :<\/p><p>\\[\\chi =7-8+3=-1+3=2\\]\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-f083bfd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f083bfd\" 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-b923cb5\" data-id=\"b923cb5\" 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-25b18bd elementor-widget elementor-widget-text-editor\" data-id=\"25b18bd\" 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>Les \u00e9l\u00e8ves ont d&rsquo;abord remarqu\u00e9 que transformer un graphe en figure ne comportant que des triangles et sans changer sa caract\u00e9ristique d&rsquo;Euler est toujours possible :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/triangulation.png\" alt=\"triangulation d'un graphe planaire\" width=\"633\" height=\"251\" \/><\/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-48cb3b8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"48cb3b8\" 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-5d4878f\" data-id=\"5d4878f\" 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-37db922 elementor-widget elementor-widget-text-editor\" data-id=\"37db922\" 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>Ils ont ensuite \u00e9tabli quelques r\u00e8gles de simplification\u00a0<span style=\"font-size: 18px; font-style: normal; font-weight: 400;\">(qu&rsquo;ils ont d\u00e9montr\u00e9s sur des exemples)<\/span><span style=\"font-size: 18px; font-style: normal; font-weight: 400;\">\u00a0<\/span><span style=\"font-size: 18px;\">\u00a0qui ne changent pas la caract\u00e9ristique d&rsquo;Euler d&rsquo;un graphe :<\/span><\/p><ul><li>\u00a0on peut supprimer certaines ar\u00eates du graphe sans changer son $\\chi $.<\/li><li><span style=\"font-size: 18px; font-style: normal; font-weight: 400;\">on peut supprimer un chapeau du graphe sans changer son $\\chi $.<\/span><\/li><\/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-90ed331 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"90ed331\" 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-bae9949\" data-id=\"bae9949\" 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-dae093b elementor-widget elementor-widget-text-editor\" data-id=\"dae093b\" 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>D<b>\u00e9monstration (sur un exemple) :<\/b> supprimer un chapeau ne change pas la caract\u00e9ristique d&rsquo;un graphe.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-18055 size-full\" src=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/supression-chapeau.png\" alt=\"suppression d'un chapeau d'un graphe planaire\" width=\"443\" height=\"156\" srcset=\"https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/supression-chapeau.png 443w, https:\/\/college-willy-ronis.fr\/maths\/wp-content\/uploads\/2024\/10\/supression-chapeau-300x106.png 300w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/>On remarque sur l&rsquo;exemple ci-dessus que retirer $n$ chapeaux reviendrait \u00e0 retirer $n$ sommets, $n$ faces et $2n$ ar\u00eates. Ainsi, en appelant $\\chi&rsquo;$ la caract\u00e9ristique d&rsquo;Euler du graphe d&rsquo;arriv\u00e9e :<br>\\[\\chi&rsquo; =(s-n)-(a-2n)+f-n=s-n-a+2n+f-n=s-a+f=\\chi\\]\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-7d5ab9d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7d5ab9d\" 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-1898612\" data-id=\"1898612\" 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-50170e6 elementor-widget elementor-widget-text-editor\" data-id=\"50170e6\" 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>En appliquant les r\u00e8gles pr\u00e9c\u00e9dentes, on peut toujours transformer un graphe planaire quelconque en un segment dont la caract\u00e9ristique d&rsquo;Euler vaut 2. Ainsi, la caract\u00e9ristique d&rsquo;Euler d&rsquo;un graphe planaire est toujours \u00e9gale \u00e0 2 et il en est de m\u00eame pour un poly\u00e8dre convexe.<\/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-8473133 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8473133\" 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-996672e\" data-id=\"996672e\" 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-de736ee elementor-button-info elementor-align-center elementor-widget elementor-widget-button\" data-id=\"de736ee\" 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\/9CFjDJKAzHHPxya\" 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\">Le sujet<\/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-70db647 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"70db647\" 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-b87b11a\" data-id=\"b87b11a\" 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-7dabb78 elementor-widget elementor-widget-spacer\" data-id=\"7dabb78\" 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>Section d&rsquo;excellence Caract\u00e9ristique d&rsquo;Euler A) Poly\u00e8dres convexes D\u00e9finition : Un poly\u00e8dre est un solide dont toutes les faces sont des polygones. D\u00e9finition : Un poly\u00e8dre est convexe si toutes ses diagonales sont enti\u00e8rement contenues dans son int\u00e9rieur. Exemple :\u00a0Un poly\u00e8dre convexe (Dod\u00e9ca\u00e8dre r\u00e9gulier). Un poly\u00e8dre non convexe (Icosa\u00e8dre de Jessen). B) Caract\u00e9ristique d&rsquo;Euler d&rsquo;un poly\u00e8dre [&hellip;]<\/p>\n","protected":false},"author":7440,"featured_media":18033,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"no-sidebar","site-content-layout":"","ast-site-content-layout":"full-width-container","site-content-style":"default","site-sidebar-style":"unboxed","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":"disabled","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":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[119],"tags":[],"class_list":["post-18001","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-actualites"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.2 - 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