{"id":7743,"date":"2022-07-05T10:05:10","date_gmt":"2022-07-05T08:05:10","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7743"},"modified":"2022-07-05T10:05:12","modified_gmt":"2022-07-05T08:05:12","slug":"demonstration-de-la-premiere-egalite-euclide","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/05\/demonstration-de-la-premiere-egalite-euclide\/","title":{"rendered":"D\u00e9monstration de la premi\u00e8re \u00e9galit\u00e9 d&rsquo;Euclide"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">D\u00e9monstration de la premi\u00e8re \u00e9galit\u00e9 d&rsquo;Euclide pas \u00e0 pas apr\u00e8s avoir vu bien entendu ce qu&rsquo;\u00e9tait cette \u00e9galit\u00e9.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/05\/demonstration-de-la-premiere-egalite-euclide\/#Lenonce\" >L&rsquo;\u00e9nonc\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/05\/demonstration-de-la-premiere-egalite-euclide\/#Demonstration_de_la_premiere_egalite_dEuclide_a_laide_du_theoreme_de_Pythagore\" >D\u00e9monstration de la premi\u00e8re \u00e9galit\u00e9 d&rsquo;Euclide: \u00e0 l&rsquo;aide du th\u00e9or\u00e8me de Pythagore<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/05\/demonstration-de-la-premiere-egalite-euclide\/#A_quel_niveau_peut-on_faire_cette_demonstration\" >\u00c0 quel niveau peut-on faire cette d\u00e9monstration ?<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Lenonce\"><\/span>L&rsquo;\u00e9nonc\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"1517\" height=\"639\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1.png\" alt=\"d\u00e9monstration premi\u00e8re \u00e9galit\u00e9 euclide\" class=\"wp-image-7750\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1.png 1517w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1-300x126.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1-600x253.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1-1024x431.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-1-768x324.png 768w\" sizes=\"auto, (max-width: 1517px) 100vw, 1517px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re un triangle ABC rectangle en C.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soit H le pied de la hauteur de ce triangle issue du sommet C.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si l&rsquo;on convient d&rsquo;utiliser les notations port\u00e9es sur le dessin ci-dessus, alors:$$h^2=pk.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demonstration_de_la_premiere_egalite_dEuclide_a_laide_du_theoreme_de_Pythagore\"><\/span>D\u00e9monstration de la premi\u00e8re \u00e9galit\u00e9 d&rsquo;Euclide: \u00e0 l&rsquo;aide du th\u00e9or\u00e8me de Pythagore<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Pythagore appliqu\u00e9 dans le triangle ABC rectangle en C,$$c^2=a^2+b^2.$$On peut alors en conclure que:$$a^2=c^2-b^2$$et donc:$$a^2=(p+k)^2-b^2=p^2+k^2+2pk-b^2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le m\u00eame th\u00e9or\u00e8me appliqu\u00e9 dans le triangle CHA rectangle en H,$$h^2=b^2-p^2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Appliqu\u00e9 dans le triangle CHB rectangle en H, on a:$$h^2=a^2-k^2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors:$$\\begin{array}{rl} b^2-p^2 &amp; = a^2-k^2\\\\b^2-p^2 &amp; = p^2+k^2+2pk-b^2-k^2 \\\\ 2b^2 &amp; = 2p^2 + 2pk\\\\ b^2 &amp; = p^2 + pk\\end{array}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi,$$h^2=b^2-p^2=p^2 + pk-p^2\\quad\\text{soit}\\quad h^2=pk.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"A_quel_niveau_peut-on_faire_cette_demonstration\"><\/span>\u00c0 quel niveau peut-on faire cette d\u00e9monstration ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comme vous pouvez le constater, pour d\u00e9montrer cette \u00e9galit\u00e9 d&rsquo;<a href=\"https:\/\/fr.wikipedia.org\/wiki\/Euclide\" target=\"_blank\" rel=\"noreferrer noopener\">Euclide<\/a>, j&rsquo;utilise l&rsquo;\u00e9galit\u00e9 de Pythagore et la notion de transitivit\u00e9. On peut donc envisager cette d\u00e9monstration d\u00e8s la classe de 4\u00e8me.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De plus, elle n&rsquo;est pas tr\u00e8s longue ni compliqu\u00e9e. En guidant les \u00e9l\u00e8ves avec quelques questions, elle permet \u00e0 ces derniers d&rsquo;acqu\u00e9rir de l&rsquo;aisance en justification. La classe de 4\u00e8me est justement le bon moment pour assoir l&rsquo;art de la d\u00e9monstration.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour les abonn\u00e9\u00b7e\u00b7s de ce site, voici le code source \\(\\LaTeX\\) de la figure (en TiKZ).<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{standalone}\n\\usepackage{tikz}\n\\usetikzlibrary{calc}\n\\setlength{\\parindent}{0pt}\n\\begin{document}\n\\begin{tikzpicture}\n\\draw (-4,0) node[left] {A} -- (4,0) node[right] {B} -- (40:4) node[midway,above,rotate=-70] {$a$} node[above right] {C} -- (-4,0) node[midway,above,rotate=30] {$b$};\n\\draw (40:4) -- ({4*cos(40)},0) node[below] {H} node[midway,left] {$h$};\n\\draw ({4*cos(40)},0.2) -- ({4*cos(40)+0.2},0.2) -- ({4*cos(40)+0.2},0);\n\\begin{scope}[shift=(40:4),rotate=-157]\n\\draw (0,0.2) -- (0.2,0.2) -- (0.2,0);\n\\end{scope}\n\\draw[&lt;->,>=latex] (-4,-.3) -- ({4*cos(40)-.1},-.3) node[midway,fill=white] {$p$};\n\\draw[&lt;->,>=latex] ({4*cos(40)+.1},-.3) -- (4,-.3) node[midway,fill=white] {$k$};\n\\draw[&lt;->,>=latex] (-4,-.5) -- (4,-.5) node[midway,fill=white,inner ysep=0pt] {$c$};\n\\end{tikzpicture}\n\\end{document}<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>D\u00e9monstration de la premi\u00e8re \u00e9galit\u00e9 d&rsquo;Euclide pas \u00e0 pas apr\u00e8s avoir vu bien entendu ce qu&rsquo;\u00e9tait cette \u00e9galit\u00e9.<\/p>\n","protected":false},"author":1,"featured_media":7757,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,6],"tags":[88,156,224],"class_list":["post-7743","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-mathematiques","tag-latex","tag-pythagore","tag-tikz"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - 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