Thal\u00e8s et le Nil : comment mesurer sans toucher \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --ocre: #c8850a;\n --ocre-clair: #f0c060;\n --rouge: #b03a2e;\n --vert: #2e6b4f;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n \/* HERO *\/\n .hero {\n background: #0a1f14;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u25b3 \u2225 \u223c \u03b8\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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Et comment cette m\u00eame id\u00e9e permet aujourd’hui de mesurer la largeur d’un fleuve au B\u00e9nin sans se mouiller les pieds.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 9 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Imagine-toi au bord de l’Ou\u00e9m\u00e9, le plus long fleuve du B\u00e9nin. Tu dois mesurer sa largeur. Tu n’as pas de bateau. Tu n’as pas de corde assez longue. Tu n’as pas d’instrument de mesure sophistiqu\u00e9. Tu n’as qu’un b\u00e2ton, tes yeux, et ta t\u00eate.\n <\/p>\n\n <p>\n Ce probl\u00e8me, en apparence impossible, a \u00e9t\u00e9 r\u00e9solu il y a plus de 2 500 ans par un homme nomm\u00e9 <strong>Thal\u00e8s de Milet<\/strong>. Et sa solution est d’une \u00e9l\u00e9gance qui coupe le souffle.\n <\/p>\n\n <h2>Qui \u00e9tait Thal\u00e8s ?<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Portrait historique<\/span>\n <p>\n Thal\u00e8s de Milet (vers 624\u2013546 av. J.-C.) est consid\u00e9r\u00e9 comme l’un des tout premiers philosophes et math\u00e9maticiens de l’histoire occidentale. Il vivait dans l’actuelle Turquie, mais voyageait beaucoup \u2014 notamment en <strong>\u00c9gypte<\/strong>, o\u00f9 il fut confront\u00e9 \u00e0 un d\u00e9fi lanc\u00e9 par les pr\u00eatres \u00e9gyptiens : mesurer la hauteur de la Grande Pyramide de Kh\u00e9ops.\n <\/p>\n <p>\n Les \u00c9gyptiens eux-m\u00eames ne savaient plus exactement quelle \u00e9tait la hauteur de leur pyramide. Thal\u00e8s, lui, trouva la r\u00e9ponse en quelques minutes \u2014 avec seulement un b\u00e2ton plant\u00e9 dans le sol et l’observation de son ombre.\n <\/p>\n <p>\n La l\u00e9gende dit que les pr\u00eatres \u00e9gyptiens, stup\u00e9faits, lui demand\u00e8rent comment il avait fait. Sa r\u00e9ponse fut simple : <em>\u00ab\u00a0J’ai observ\u00e9 ce que vous n’avez jamais pens\u00e9 \u00e0 regarder.\u00a0\u00bb<\/em>\n <\/p>\n <\/div>\n\n <h2>L’id\u00e9e g\u00e9niale : les ombres parlent<\/h2>\n\n <p>\n Voici ce que Thal\u00e8s a compris \u2014 et c’est d’une simplicit\u00e9 d\u00e9sarmante une fois qu’on le voit.\n <\/p>\n\n <p>\n Un jour, \u00e0 une heure pr\u00e9cise, Thal\u00e8s plante un b\u00e2ton vertical dans le sol. Il mesure la longueur de l’ombre du b\u00e2ton. Il mesure aussi la hauteur du b\u00e2ton. Il observe alors quelque chose de remarquable :\n <\/p>\n\n <div class=\"formule\">\n ombre du b\u00e2ton \/ hauteur du b\u00e2ton = ombre de la pyramide \/ hauteur de la pyramide\n <\/div>\n\n <p>\n Autrement dit, \u00e0 la m\u00eame heure et au m\u00eame endroit, <strong>tous les objets verticaux ont des ombres proportionnelles \u00e0 leur hauteur<\/strong>. Le soleil est tellement loin que ses rayons arrivent pratiquement parall\u00e8les sur Terre. Donc les triangles form\u00e9s par chaque objet et son ombre sont tous <em>semblables<\/em>.\n <\/p>\n\n <div class=\"definition\">\n <strong>Triangles semblables :<\/strong> Deux triangles sont semblables s’ils ont les m\u00eames angles. Leurs c\u00f4t\u00e9s sont alors proportionnels \u2014 dans le m\u00eame rapport.\n <\/div>\n\n <h2>Le sch\u00e9ma qui explique tout<\/h2>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 Thal\u00e8s et les ombres \u2014 sch\u00e9ma illustratif<\/div>\n <svg width=\"100%\" viewBox=\"0 0 600 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n \n <line x1=\"20\" y1=\"230\" x2=\"580\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"2\"\/>\n\n \n <line x1=\"100\" y1=\"230\" x2=\"100\" y2=\"150\" stroke=\"#0f0e0b\" stroke-width=\"3\"\/>\n <text x=\"75\" y=\"195\" font-size=\"12\" fill=\"#6b6560\">b\u00e2ton<\/text>\n \n <line x1=\"100\" y1=\"230\" x2=\"160\" y2=\"230\" stroke=\"#c8850a\" stroke-width=\"3\"\/>\n <text x=\"115\" y=\"248\" font-size=\"11\" fill=\"#c8850a\">ombre b.<\/text>\n\n \n <line x1=\"100\" y1=\"150\" x2=\"160\" y2=\"230\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n\n \n <polygon points=\"340,230 460,230 400,80\" fill=\"none\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <text x=\"380\" y=\"260\" font-size=\"12\" fill=\"#6b6560\">pyramide<\/text>\n \n <line x1=\"460\" y1=\"230\" x2=\"560\" y2=\"230\" stroke=\"#c8850a\" stroke-width=\"3\"\/>\n <text x=\"490\" y=\"248\" font-size=\"11\" fill=\"#c8850a\">ombre p.<\/text>\n\n \n <line x1=\"400\" y1=\"80\" x2=\"560\" y2=\"230\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n\n \n <text x=\"152\" y=\"222\" font-size=\"14\" fill=\"#2e6b4f\">\u03b1<\/text>\n <text x=\"548\" y=\"222\" font-size=\"14\" fill=\"#2e6b4f\">\u03b1<\/text>\n\n \n <line x1=\"88\" y1=\"150\" x2=\"88\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"1.5\" stroke-dasharray=\"4,2\"\/>\n <text x=\"60\" y=\"148\" font-size=\"11\" fill=\"#2e6b4f\">h\u2081<\/text>\n\n <line x1=\"388\" y1=\"80\" x2=\"388\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"1.5\" stroke-dasharray=\"4,2\"\/>\n <text x=\"362\" y=\"78\" font-size=\"11\" fill=\"#2e6b4f\">H = ?<\/text>\n\n \n <line x1=\"30\" y1=\"20\" x2=\"60\" y2=\"20\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <text x=\"65\" y=\"24\" font-size=\"11\" fill=\"#6b6560\">Rayons du soleil (parall\u00e8les)<\/text>\n <text x=\"65\" y=\"40\" font-size=\"11\" fill=\"#2e6b4f\">\u03b1 = angles \u00e9gaux<\/text>\n <\/svg>\n <\/div>\n\n <p>\n Les deux triangles \u2014 celui du b\u00e2ton avec son ombre, et celui de la pyramide avec son ombre \u2014 ont le <strong>m\u00eame angle<\/strong> avec le sol (l’angle form\u00e9 par les rayons du soleil). Ils sont donc semblables. Et des triangles semblables ont des c\u00f4t\u00e9s proportionnels.\n <\/p>\n\n <h2>Le calcul concret<\/h2>\n\n <p>Voici comment Thal\u00e8s a calcul\u00e9 la hauteur de la pyramide :<\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcca Donn\u00e9es mesur\u00e9es par Thal\u00e8s<\/span>\n <p><strong>Hauteur du b\u00e2ton (h\u2081) :<\/strong> 2 m\u00e8tres<\/p>\n <p><strong>Longueur de l’ombre du b\u00e2ton (o\u2081) :<\/strong> 3 m\u00e8tres<\/p>\n <p><strong>Longueur de l’ombre de la pyramide (o\u2082) :<\/strong> 180 m\u00e8tres<\/p>\n <p><strong>Hauteur de la pyramide (H) :<\/strong> inconnue<\/p>\n <\/div>\n\n <p>Par le th\u00e9or\u00e8me de Thal\u00e8s (proportionnalit\u00e9) :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> h\u2081 \/ o\u2081 = H \/ o\u2082<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> 2 \/ 3 = H \/ 180<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> H = (2 \u00d7 180) \/ 3<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> H = 360 \/ 3 = <strong>120 m\u00e8tres<\/strong><\/div>\n\n <div class=\"formule\">H = 120 m\u00e8tres \u2713<\/div>\n\n <p>\n La hauteur r\u00e9elle de la Grande Pyramide de Kh\u00e9ops est de 138 m\u00e8tres (\u00e0 l’origine). Les chiffres exacts varient selon les sources historiques, mais la m\u00e9thode, elle, est parfaitement exacte.\n <\/p>\n\n <h2>Et au B\u00e9nin ? Mesurer l’Ou\u00e9m\u00e9 sans se mouiller<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Application africaine<\/span>\n <p>\n Tu es au bord de l’Ou\u00e9m\u00e9. En face, tu vois un arbre (point A). Tu plantes un piquet l\u00e0 o\u00f9 tu te trouves (point B). Tu marches le long de la rive sur une distance connue, disons 50 m\u00e8tres, et tu plantes un second piquet (point C). Depuis C, tu t’\u00e9loignes perpendiculairement \u00e0 la rive jusqu’au point D, d’o\u00f9 tu vois l’arbre A et le piquet B parfaitement align\u00e9s.\n <\/p>\n <p>\n Le th\u00e9or\u00e8me de Thal\u00e8s te donne alors la largeur du fleuve AB \u2014 sans jamais traverser l’eau. C’est exactement ainsi que les ing\u00e9nieurs africains et les g\u00e9om\u00e8tres traditionnels mesuraient les \u00e9tendues inaccessibles.\n <\/p>\n <\/div>\n\n <h2>Le th\u00e9or\u00e8me de Thal\u00e8s \u2014 l’\u00e9nonc\u00e9 officiel<\/h2>\n\n <div class=\"definition\">\n <strong>Th\u00e9or\u00e8me de Thal\u00e8s :<\/strong> Si deux droites sont coup\u00e9es par des parall\u00e8les, alors les rapports des segments correspondants sont \u00e9gaux. Autrement dit, si D\u2081 \u2225 D\u2082 \u2225 D\u2083, alors les segments qu’elles d\u00e9coupent sur deux s\u00e9cantes quelconques sont proportionnels.\n <\/div>\n\n <blockquote>\n La g\u00e9om\u00e9trie, c’est l’art de raisonner juste sur des figures fausses.\n <cite>\u2014 Henri Poincar\u00e9, math\u00e9maticien fran\u00e7ais<\/cite>\n <\/blockquote>\n\n <p>\n Ce que voulait dire Poincar\u00e9 : peu importe si ton sch\u00e9ma n’est pas parfaitement dessin\u00e9. Ce qui compte, c’est la logique du raisonnement. Thal\u00e8s l’avait compris 25 si\u00e8cles avant lui.\n <\/p>\n\n <h2>Pourquoi c’est magique<\/h2>\n\n <p>\n Ce qui est extraordinaire dans la d\u00e9marche de Thal\u00e8s, c’est qu’elle transforme un probl\u00e8me <em>physiquement impossible<\/em> \u2014 toucher le sommet d’une pyramide \u2014 en un probl\u00e8me <em>math\u00e9matiquement simple<\/em>. Il suffit de mesurer ce qui est accessible (les ombres au sol) pour d\u00e9duire ce qui ne l’est pas (la hauteur).\n <\/p>\n\n <p>\n C’est l’essence m\u00eame des math\u00e9matiques : <strong>contourner les impossibilit\u00e9s par le raisonnement<\/strong>. Cette id\u00e9e traverse toute l’histoire des sciences \u2014 de Thal\u00e8s jusqu’aux satellites GPS qui calculent ta position aujourd’hui avec des triangles dans l’espace.\n <\/p>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Au bord du lac Nokou\u00e9 au B\u00e9nin, tu veux mesurer la hauteur d’un grand fromager (arbre) de l’autre c\u00f4t\u00e9. Tu plantes un b\u00e2ton de <strong>1,5 m<\/strong> dont l’ombre mesure <strong>2 m<\/strong>. Au m\u00eame moment, l’ombre de l’arbre mesure <strong>16 m<\/strong>.\n <\/p>\n <p><strong>Question :<\/strong> Quelle est la hauteur de l’arbre ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p>Par proportionnalit\u00e9 (th\u00e9or\u00e8me de Thal\u00e8s) :<\/p>\n <p>hauteur b\u00e2ton \/ ombre b\u00e2ton = hauteur arbre \/ ombre arbre<\/p>\n <p>1,5 \/ 2 = H \/ 16<\/p>\n <p>H = (1,5 \u00d7 16) \/ 2 = 24 \/ 2 = <strong>12 m\u00e8tres<\/strong><\/p>\n <p>\u2705 L’arbre mesure <strong>12 m\u00e8tres<\/strong> de hauteur \u2014 sans jamais traverser le lac !<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>\u00c0 retenir<\/h2>\n <p>\n Thal\u00e8s nous a appris que les math\u00e9matiques sont un outil de <strong>libert\u00e9<\/strong>. Quand le monde physique nous bloque \u2014 un fleuve trop large, une pyramide trop haute, une \u00e9toile trop lointaine \u2014 la g\u00e9om\u00e9trie ouvre une porte de c\u00f4t\u00e9. Elle nous permet de mesurer l’inaccessible, de toucher l’intouchable, de voir l’invisible.\n <\/p>\n <p>\n La prochaine fois que tu regardes ton ombre au soleil, souviens-toi : tu portes en toi le m\u00eame outil que Thal\u00e8s. Il suffit de savoir l’utiliser.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans un myst\u00e8re que les abeilles ont r\u00e9solu bien avant les math\u00e9maticiens : <em>pourquoi construisent-elles des hexagones ?<\/em> Une histoire de g\u00e9om\u00e9trie, d’efficacit\u00e9 et de nature intelligente.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Thal\u00e8s et le Nil : comment mesurer sans toucher \u2014 MathsVivantes G\u00e9om\u00e9trie \u00b7 Article #2 Thal\u00e8s et le Nil :mesurer […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"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":"","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-4)","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-4)","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-4)","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":[8],"tags":[],"class_list":["post-55","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Thal\u00e8s et le Nil : comment mesurer sans toucher - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comment Thal\u00e8s a mesur\u00e9 les pyramides d'Egypte avec les ombres et un b\u00e2ton . 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top: -10px;\n font-size: 3.5rem;\n color: var(--vert);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite {\n display: block;\n font-size: 0.82rem;\n font-style: normal;\n letter-spacing: 0.1em;\n text-transform: uppercase;\n color: var(--gris);\n margin-top: 8px;\n }\n\n .conclusion {\n border-top: 2px solid var(--noir);\n margin-top: 3em;\n padding-top: 2em;\n }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n }\n<\/style>\n<\/head>\n<body>\n\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">G\u00e9om\u00e9trie \u00b7 Article #2<\/div>\n <h1>Thal\u00e8s et le Nil :<br><em>mesurer sans toucher<\/em><\/h1>\n <p class=\"hero-intro\">Comment un Grec du VI<sup>e<\/sup> si\u00e8cle avant J.-C. a-t-il mesur\u00e9 la hauteur des pyramides d’\u00c9gypte avec seulement un b\u00e2ton et son ombre ? Et comment cette m\u00eame id\u00e9e permet aujourd’hui de mesurer la largeur d’un fleuve au B\u00e9nin sans se mouiller les pieds.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 9 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Imagine-toi au bord de l’Ou\u00e9m\u00e9, le plus long fleuve du B\u00e9nin. Tu dois mesurer sa largeur. Tu n’as pas de bateau. Tu n’as pas de corde assez longue. Tu n’as pas d’instrument de mesure sophistiqu\u00e9. Tu n’as qu’un b\u00e2ton, tes yeux, et ta t\u00eate.\n <\/p>\n\n <p>\n Ce probl\u00e8me, en apparence impossible, a \u00e9t\u00e9 r\u00e9solu il y a plus de 2 500 ans par un homme nomm\u00e9 <strong>Thal\u00e8s de Milet<\/strong>. Et sa solution est d’une \u00e9l\u00e9gance qui coupe le souffle.\n <\/p>\n\n <h2>Qui \u00e9tait Thal\u00e8s ?<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Portrait historique<\/span>\n <p>\n Thal\u00e8s de Milet (vers 624\u2013546 av. J.-C.) est consid\u00e9r\u00e9 comme l’un des tout premiers philosophes et math\u00e9maticiens de l’histoire occidentale. Il vivait dans l’actuelle Turquie, mais voyageait beaucoup \u2014 notamment en <strong>\u00c9gypte<\/strong>, o\u00f9 il fut confront\u00e9 \u00e0 un d\u00e9fi lanc\u00e9 par les pr\u00eatres \u00e9gyptiens : mesurer la hauteur de la Grande Pyramide de Kh\u00e9ops.\n <\/p>\n <p>\n Les \u00c9gyptiens eux-m\u00eames ne savaient plus exactement quelle \u00e9tait la hauteur de leur pyramide. Thal\u00e8s, lui, trouva la r\u00e9ponse en quelques minutes \u2014 avec seulement un b\u00e2ton plant\u00e9 dans le sol et l’observation de son ombre.\n <\/p>\n <p>\n La l\u00e9gende dit que les pr\u00eatres \u00e9gyptiens, stup\u00e9faits, lui demand\u00e8rent comment il avait fait. Sa r\u00e9ponse fut simple : <em>\u00ab\u00a0J’ai observ\u00e9 ce que vous n’avez jamais pens\u00e9 \u00e0 regarder.\u00a0\u00bb<\/em>\n <\/p>\n <\/div>\n\n <h2>L’id\u00e9e g\u00e9niale : les ombres parlent<\/h2>\n\n <p>\n Voici ce que Thal\u00e8s a compris \u2014 et c’est d’une simplicit\u00e9 d\u00e9sarmante une fois qu’on le voit.\n <\/p>\n\n <p>\n Un jour, \u00e0 une heure pr\u00e9cise, Thal\u00e8s plante un b\u00e2ton vertical dans le sol. Il mesure la longueur de l’ombre du b\u00e2ton. Il mesure aussi la hauteur du b\u00e2ton. Il observe alors quelque chose de remarquable :\n <\/p>\n\n <div class=\"formule\">\n ombre du b\u00e2ton \/ hauteur du b\u00e2ton = ombre de la pyramide \/ hauteur de la pyramide\n <\/div>\n\n <p>\n Autrement dit, \u00e0 la m\u00eame heure et au m\u00eame endroit, <strong>tous les objets verticaux ont des ombres proportionnelles \u00e0 leur hauteur<\/strong>. Le soleil est tellement loin que ses rayons arrivent pratiquement parall\u00e8les sur Terre. Donc les triangles form\u00e9s par chaque objet et son ombre sont tous <em>semblables<\/em>.\n <\/p>\n\n <div class=\"definition\">\n <strong>Triangles semblables :<\/strong> Deux triangles sont semblables s’ils ont les m\u00eames angles. Leurs c\u00f4t\u00e9s sont alors proportionnels \u2014 dans le m\u00eame rapport.\n <\/div>\n\n <h2>Le sch\u00e9ma qui explique tout<\/h2>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 Thal\u00e8s et les ombres \u2014 sch\u00e9ma illustratif<\/div>\n <svg width=\"100%\" viewBox=\"0 0 600 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n \n <line x1=\"20\" y1=\"230\" x2=\"580\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"2\"\/>\n\n \n <line x1=\"100\" y1=\"230\" x2=\"100\" y2=\"150\" stroke=\"#0f0e0b\" stroke-width=\"3\"\/>\n <text x=\"75\" y=\"195\" font-size=\"12\" fill=\"#6b6560\">b\u00e2ton<\/text>\n \n <line x1=\"100\" y1=\"230\" x2=\"160\" y2=\"230\" stroke=\"#c8850a\" stroke-width=\"3\"\/>\n <text x=\"115\" y=\"248\" font-size=\"11\" fill=\"#c8850a\">ombre b.<\/text>\n\n \n <line x1=\"100\" y1=\"150\" x2=\"160\" y2=\"230\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n\n \n <polygon points=\"340,230 460,230 400,80\" fill=\"none\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <text x=\"380\" y=\"260\" font-size=\"12\" fill=\"#6b6560\">pyramide<\/text>\n \n <line x1=\"460\" y1=\"230\" x2=\"560\" y2=\"230\" stroke=\"#c8850a\" stroke-width=\"3\"\/>\n <text x=\"490\" y=\"248\" font-size=\"11\" fill=\"#c8850a\">ombre p.<\/text>\n\n \n <line x1=\"400\" y1=\"80\" x2=\"560\" y2=\"230\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n\n \n <text x=\"152\" y=\"222\" font-size=\"14\" fill=\"#2e6b4f\">\u03b1<\/text>\n <text x=\"548\" y=\"222\" font-size=\"14\" fill=\"#2e6b4f\">\u03b1<\/text>\n\n \n <line x1=\"88\" y1=\"150\" x2=\"88\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"1.5\" stroke-dasharray=\"4,2\"\/>\n <text x=\"60\" y=\"148\" font-size=\"11\" fill=\"#2e6b4f\">h\u2081<\/text>\n\n <line x1=\"388\" y1=\"80\" x2=\"388\" y2=\"230\" stroke=\"#2e6b4f\" stroke-width=\"1.5\" stroke-dasharray=\"4,2\"\/>\n <text x=\"362\" y=\"78\" font-size=\"11\" fill=\"#2e6b4f\">H = ?<\/text>\n\n \n <line x1=\"30\" y1=\"20\" x2=\"60\" y2=\"20\" stroke=\"#f0c060\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <text x=\"65\" y=\"24\" font-size=\"11\" fill=\"#6b6560\">Rayons du soleil (parall\u00e8les)<\/text>\n <text x=\"65\" y=\"40\" font-size=\"11\" fill=\"#2e6b4f\">\u03b1 = angles \u00e9gaux<\/text>\n <\/svg>\n <\/div>\n\n <p>\n Les deux triangles \u2014 celui du b\u00e2ton avec son ombre, et celui de la pyramide avec son ombre \u2014 ont le <strong>m\u00eame angle<\/strong> avec le sol (l’angle form\u00e9 par les rayons du soleil). Ils sont donc semblables. Et des triangles semblables ont des c\u00f4t\u00e9s proportionnels.\n <\/p>\n\n <h2>Le calcul concret<\/h2>\n\n <p>Voici comment Thal\u00e8s a calcul\u00e9 la hauteur de la pyramide :<\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcca Donn\u00e9es mesur\u00e9es par Thal\u00e8s<\/span>\n <p><strong>Hauteur du b\u00e2ton (h\u2081) :<\/strong> 2 m\u00e8tres<\/p>\n <p><strong>Longueur de l’ombre du b\u00e2ton (o\u2081) :<\/strong> 3 m\u00e8tres<\/p>\n <p><strong>Longueur de l’ombre de la pyramide (o\u2082) :<\/strong> 180 m\u00e8tres<\/p>\n <p><strong>Hauteur de la pyramide (H) :<\/strong> inconnue<\/p>\n <\/div>\n\n <p>Par le th\u00e9or\u00e8me de Thal\u00e8s (proportionnalit\u00e9) :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> h\u2081 \/ o\u2081 = H \/ o\u2082<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> 2 \/ 3 = H \/ 180<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> H = (2 \u00d7 180) \/ 3<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> H = 360 \/ 3 = <strong>120 m\u00e8tres<\/strong><\/div>\n\n <div class=\"formule\">H = 120 m\u00e8tres \u2713<\/div>\n\n <p>\n La hauteur r\u00e9elle de la Grande Pyramide de Kh\u00e9ops est de 138 m\u00e8tres (\u00e0 l’origine). Les chiffres exacts varient selon les sources historiques, mais la m\u00e9thode, elle, est parfaitement exacte.\n <\/p>\n\n <h2>Et au B\u00e9nin ? Mesurer l’Ou\u00e9m\u00e9 sans se mouiller<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Application africaine<\/span>\n <p>\n Tu es au bord de l’Ou\u00e9m\u00e9. En face, tu vois un arbre (point A). Tu plantes un piquet l\u00e0 o\u00f9 tu te trouves (point B). Tu marches le long de la rive sur une distance connue, disons 50 m\u00e8tres, et tu plantes un second piquet (point C). Depuis C, tu t’\u00e9loignes perpendiculairement \u00e0 la rive jusqu’au point D, d’o\u00f9 tu vois l’arbre A et le piquet B parfaitement align\u00e9s.\n <\/p>\n <p>\n Le th\u00e9or\u00e8me de Thal\u00e8s te donne alors la largeur du fleuve AB \u2014 sans jamais traverser l’eau. C’est exactement ainsi que les ing\u00e9nieurs africains et les g\u00e9om\u00e8tres traditionnels mesuraient les \u00e9tendues inaccessibles.\n <\/p>\n <\/div>\n\n <h2>Le th\u00e9or\u00e8me de Thal\u00e8s \u2014 l’\u00e9nonc\u00e9 officiel<\/h2>\n\n <div class=\"definition\">\n <strong>Th\u00e9or\u00e8me de Thal\u00e8s :<\/strong> Si deux droites sont coup\u00e9es par des parall\u00e8les, alors les rapports des segments correspondants sont \u00e9gaux. Autrement dit, si D\u2081 \u2225 D\u2082 \u2225 D\u2083, alors les segments qu’elles d\u00e9coupent sur deux s\u00e9cantes quelconques sont proportionnels.\n <\/div>\n\n <blockquote>\n La g\u00e9om\u00e9trie, c’est l’art de raisonner juste sur des figures fausses.\n <cite>\u2014 Henri Poincar\u00e9, math\u00e9maticien fran\u00e7ais<\/cite>\n <\/blockquote>\n\n <p>\n Ce que voulait dire Poincar\u00e9 : peu importe si ton sch\u00e9ma n’est pas parfaitement dessin\u00e9. Ce qui compte, c’est la logique du raisonnement. Thal\u00e8s l’avait compris 25 si\u00e8cles avant lui.\n <\/p>\n\n <h2>Pourquoi c’est magique<\/h2>\n\n <p>\n Ce qui est extraordinaire dans la d\u00e9marche de Thal\u00e8s, c’est qu’elle transforme un probl\u00e8me <em>physiquement impossible<\/em> \u2014 toucher le sommet d’une pyramide \u2014 en un probl\u00e8me <em>math\u00e9matiquement simple<\/em>. Il suffit de mesurer ce qui est accessible (les ombres au sol) pour d\u00e9duire ce qui ne l’est pas (la hauteur).\n <\/p>\n\n <p>\n C’est l’essence m\u00eame des math\u00e9matiques : <strong>contourner les impossibilit\u00e9s par le raisonnement<\/strong>. Cette id\u00e9e traverse toute l’histoire des sciences \u2014 de Thal\u00e8s jusqu’aux satellites GPS qui calculent ta position aujourd’hui avec des triangles dans l’espace.\n <\/p>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Au bord du lac Nokou\u00e9 au B\u00e9nin, tu veux mesurer la hauteur d’un grand fromager (arbre) de l’autre c\u00f4t\u00e9. Tu plantes un b\u00e2ton de <strong>1,5 m<\/strong> dont l’ombre mesure <strong>2 m<\/strong>. Au m\u00eame moment, l’ombre de l’arbre mesure <strong>16 m<\/strong>.\n <\/p>\n <p><strong>Question :<\/strong> Quelle est la hauteur de l’arbre ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p>Par proportionnalit\u00e9 (th\u00e9or\u00e8me de Thal\u00e8s) :<\/p>\n <p>hauteur b\u00e2ton \/ ombre b\u00e2ton = hauteur arbre \/ ombre arbre<\/p>\n <p>1,5 \/ 2 = H \/ 16<\/p>\n <p>H = (1,5 \u00d7 16) \/ 2 = 24 \/ 2 = <strong>12 m\u00e8tres<\/strong><\/p>\n <p>\u2705 L’arbre mesure <strong>12 m\u00e8tres<\/strong> de hauteur \u2014 sans jamais traverser le lac !<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>\u00c0 retenir<\/h2>\n <p>\n Thal\u00e8s nous a appris que les math\u00e9matiques sont un outil de <strong>libert\u00e9<\/strong>. Quand le monde physique nous bloque \u2014 un fleuve trop large, une pyramide trop haute, une \u00e9toile trop lointaine \u2014 la g\u00e9om\u00e9trie ouvre une porte de c\u00f4t\u00e9. Elle nous permet de mesurer l’inaccessible, de toucher l’intouchable, de voir l’invisible.\n <\/p>\n <p>\n La prochaine fois que tu regardes ton ombre au soleil, souviens-toi : tu portes en toi le m\u00eame outil que Thal\u00e8s. Il suffit de savoir l’utiliser.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans un myst\u00e8re que les abeilles ont r\u00e9solu bien avant les math\u00e9maticiens : <em>pourquoi construisent-elles des hexagones ?<\/em> Une histoire de g\u00e9om\u00e9trie, d’efficacit\u00e9 et de nature intelligente.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Thal\u00e8s et le Nil : comment mesurer sans toucher \u2014 MathsVivantes G\u00e9om\u00e9trie \u00b7 Article #2 Thal\u00e8s et le Nil :mesurer […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"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":"","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-4)","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-4)","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-4)","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":[8],"tags":[],"class_list":["post-55","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Thal\u00e8s et le Nil : comment mesurer sans toucher - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comment Thal\u00e8s a mesur\u00e9 les pyramides d'Egypte avec les ombres et un b\u00e2ton . 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