{"id":8177,"date":"2023-02-15T15:54:36","date_gmt":"2023-02-15T14:54:36","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=8177"},"modified":"2023-02-20T14:54:31","modified_gmt":"2023-02-20T13:54:31","slug":"tangentes-a-un-cercle-et-droites-concourantes","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/","title":{"rendered":"Tangentes \u00e0 un cercle et droites concourantes"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je vais ici vous proposer un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel \u00e0 un r\u00e9sultats math\u00e9matique peu connu: le th\u00e9or\u00e8me de Ceva.<\/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\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Tangentes_a_un_cercle_et_droites_concourantes_lenonce\" >Tangentes \u00e0 un cercle et droites concourantes: l&rsquo;\u00e9nonc\u00e9<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Tangentes_a_un_cercle_et_droites_concourantes_une_figure\" >Tangentes \u00e0 un cercle et droites concourantes: une figure<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Tangentes_a_un_cercle_et_droites_concourantes_le_fichier_geogebra\" >Tangentes \u00e0 un cercle et droites concourantes: le fichier geogebra<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Tangentes_a_un_cercle_et_droites_concourantes_une_solution\" >Tangentes \u00e0 un cercle et droites concourantes: une solution<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Preliminaires\" >Pr\u00e9liminaires<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Notion_de_barycentres\" >Notion de barycentres<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Associativite_du_barycentre\" >Associativit\u00e9 du barycentre<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Retour_au_probleme\" >Retour au probl\u00e8me<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#Theoreme_de_Ceva\" >Th\u00e9or\u00e8me de Ceva<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tangentes_a_un_cercle_et_droites_concourantes_lenonce\"><\/span>Tangentes \u00e0 un cercle et droites concourantes: l&rsquo;\u00e9nonc\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On consid\u00e8re un cercle \\(\\mathcal{C}\\) de centre O, et trois points A, B, C sur ce cercle, non diam\u00e9tralement oppos\u00e9s deux \u00e0 deux. Autrement dit, [AB], [AC] et [BC] ne sont pas des diam\u00e8tres de \\(\\mathcal{C}\\).<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On trace alors les tangentes \\(T_A\\), \\(T_B\\) et \\(T_C\\) \u00e0 \\(\\mathcal{C}\\) respectivement en A, B et C.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">\\(T_A\\) et \\(T_B\\) se coupent en C&rsquo;,<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">\\(T_A\\) et \\(T_C\\) se coupent en B&rsquo;,<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">\\(T_B\\) et \\(T_C\\) se coupent en A&rsquo;.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Montrer que (AA&rsquo;), (BB&rsquo;) et (CC&rsquo;) sont concourantes, c&rsquo;est-\u00e0-dire qu&rsquo;elles se coupent en un unique point.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Tous les niveaux de r\u00e9solution sont accept\u00e9s.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Vous pourrez proposer votre solution en commentaire. Celui-ci ne sera pas publi\u00e9 s&rsquo;il comporte des erreurs math\u00e9matiques. En revanche, s&rsquo;il est correct, il apparaitra dans la section suivante de fa\u00e7on anonyme si vous le souhaitez (dans ce cas, merci de le pr\u00e9ciser dans votre commentaire).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tangentes_a_un_cercle_et_droites_concourantes_une_figure\"><\/span>Tangentes \u00e0 un cercle et droites concourantes: une figure<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"804\" height=\"432\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image.png\" alt=\"tangentes cercle droites concourantes\" class=\"wp-image-8178\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image.png 804w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-600x322.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-300x161.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-768x413.png 768w\" sizes=\"auto, (max-width: 804px) 100vw, 804px\" \/><\/a><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tangentes_a_un_cercle_et_droites_concourantes_le_fichier_geogebra\"><\/span>Tangentes \u00e0 un cercle et droites concourantes: le fichier geogebra<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-22f92cab-a95b-4658-a821-94adb8d604b4\" href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/tangentes.ggb\">tangentes<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/tangentes.ggb\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-22f92cab-a95b-4658-a821-94adb8d604b4\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tangentes_a_un_cercle_et_droites_concourantes_une_solution\"><\/span>Tangentes \u00e0 un cercle et droites concourantes: une solution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Preliminaires\"><\/span>Pr\u00e9liminaires<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Par construction, les triangles OAC&rsquo; et OBC&rsquo; sont rectangles respectivement en A et B, ont un c\u00f4t\u00e9 en commun, le segment [OC&rsquo;], et deux c\u00f4t\u00e9s de m\u00eame mesure, \u00e0 savoir OA = OB. Ainsi, ils sont semblables et on peut alors conclure que AC&rsquo; = BC&rsquo;.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Par un raisonnement analogue, on peut d\u00e9montrer que CA&rsquo; = BA&rsquo; et CB&rsquo; = AB&rsquo;.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Notion_de_barycentres\"><\/span>Notion de barycentres<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Notons:$$\\alpha=AC&rsquo;=BC&rsquo;\\quad,\\quad \\beta=BA&rsquo;=CA&rsquo;\\quad,\\quad\\gamma=CB&rsquo;=AB&rsquo;.$$<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"745\" height=\"382\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-1.png\" alt=\"\" class=\"wp-image-8185\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-1.png 745w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-1-600x308.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/image-1-300x154.png 300w\" sizes=\"auto, (max-width: 745px) 100vw, 745px\" \/><figcaption class=\"wp-element-caption\">Avec pond\u00e9rations<\/figcaption><\/figure>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut \u00e9crire:$$\\gamma \\vec{AC&rsquo;} + \\alpha\\vec{AB&rsquo;}=\\vec{0}.$$On dit alors que A est le <em><strong>barycentre<\/strong><\/em> du syst\u00e8me pond\u00e9r\u00e9 \\( \\{(B&rsquo;;\\alpha),(C&rsquo;;\\gamma)\\}\\), et on peut noter:$$A=\\text{bar}\\{ (B&rsquo;;\\alpha),(C&rsquo;;\\gamma) \\}.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En multipliant l&rsquo;\u00e9galit\u00e9 vectorielle par \\(\\beta\\), on a:$$\\beta\\gamma \\vec{AC&rsquo;} + \\alpha\\beta\\vec{AB&rsquo;}=\\vec{0}.$$Autrement dit:$$A=\\text{bar}\\{ (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\}.\\quad\\quad(1)$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut alors d\u00e9montrer de m\u00eame que:$$B=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma),(C&rsquo;;\\beta\\gamma) \\}\\quad\\quad(2)$$et$$C=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma),(B&rsquo;;\\alpha\\beta) \\}.\\quad\\quad(3)$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Consid\u00e9rons maintenant le point I d\u00e9finit ainsi:$$I=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma), (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\}.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Associativite_du_barycentre\"><\/span>Associativit\u00e9 du barycentre<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Posons:$$N=\\text{bar}\\{ (P;a),(Q;b) \\}$$avec\\(a+b\\neq0\\). Alors,$$a\\vec{NP}+b\\vec{NQ}=\\vec{0}.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Posons maintenant $$M=\\text{bar}\\{ (P;a),(Q;b),(R,c) \\}$$Alors:$$\\begin{array}{ll} &amp; a\\vec{MP}+b\\vec{MQ}+c\\vec{MR}=\\vec{0}\\\\\\iff &amp; a\\big(\\vec{MN}+\\vec{NP}\\big)+b\\big(\\vec{MN}+\\vec{NQ}\\big)+c\\vec{MR}=\\vec{0}\\\\\\iff &amp; (a+b)\\vec{MN}+\\underbrace{a\\vec{NP}+b\\vec{NQ}}_{=\\vec{0}}+c\\vec{MR}=\\vec{0}\\\\\\iff &amp; (a+b)\\vec{MN}+c\\vec{MR}=\\vec{0}\\\\\\iff &amp; M=\\text{bar}\\{ (N;a+b),(R;c) \\} \\end{array} $$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ce dernier r\u00e9sultat est appel\u00e9 l&rsquo;<em><strong>associativit\u00e9 du barycentre<\/strong><\/em>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Retour_au_probleme\"><\/span>Retour au probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On a:$$\\left.\\begin{array}{l}I=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma), (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\}\\\\A=\\text{bar}\\{ (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\} \\end{array}\\right\\}\\Rightarrow I=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma),(A;\\alpha\\beta+\\beta\\gamma) \\}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut alors dire que:$$I\\in(AA&rsquo;).$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">De m\u00eame,$$\\left.\\begin{array}{l}I=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma), (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\}\\\\B=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma),(C&rsquo;;\\beta\\gamma) \\} \\end{array}\\right\\}\\Rightarrow I=\\text{bar}\\{ (B&rsquo;;\\alpha\\beta),(B;\\alpha\\gamma+\\beta\\gamma) \\}$$ et $$\\left.\\begin{array}{l}I=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma), (B&rsquo;;\\alpha\\beta),(C&rsquo;;\\beta\\gamma) \\}\\\\C=\\text{bar}\\{ (A&rsquo;;\\alpha\\gamma),(B&rsquo;;\\alpha\\beta) \\}\\end{array}\\right\\}\\Rightarrow I=\\text{bar}\\{ (C&rsquo;;\\beta\\gamma),(C;\\alpha\\beta+\\alpha\\gamma) \\}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On en conclut alors que \\( I\\in(BB&rsquo;)\\) et \\(I\\in(CC&rsquo;)\\).<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Par cons\u00e9quent, I est le point d&rsquo;intersection des droites (AA&rsquo;), (BB&rsquo;) et (CC&rsquo;).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Theoreme_de_Ceva\"><\/span>Th\u00e9or\u00e8me de Ceva<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ce que nous venons de d\u00e9montrer est un cas particulier du <em>th\u00e9or\u00e8me de Ceva<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Vous en saurez plus sur ce th\u00e9or\u00e8me en regardant la page <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Th%C3%A9or%C3%A8me_de_Ceva\" target=\"_blank\" rel=\"noreferrer noopener\">wikipedia<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Je vais ici vous proposer un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel \u00e0 un r\u00e9sultats math\u00e9matique peu connu: le th\u00e9or\u00e8me de Ceva.<\/p>\n","protected":false},"author":1,"featured_media":8182,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[380,60,378,379],"class_list":["post-8177","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-barycentres","tag-cercle","tag-ceva-theoreme","tag-droites-concourantes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2023-02-15T14:54:36+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-02-20T13:54:31+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"740\" \/>\n\t<meta property=\"og:image:height\" content=\"198\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Tangentes \u00e0 un cercle et droites concourantes\",\"datePublished\":\"2023-02-15T14:54:36+00:00\",\"dateModified\":\"2023-02-20T13:54:31+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/\"},\"wordCount\":771,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg\",\"keywords\":[\"barycentres\",\"cercle\",\"Ceva (th\u00e9or\u00e8me)\",\"droites concourantes\"],\"articleSection\":[\"Math\u00e9matiques\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/\",\"name\":\"Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg\",\"datePublished\":\"2023-02-15T14:54:36+00:00\",\"dateModified\":\"2023-02-20T13:54:31+00:00\",\"description\":\"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg\",\"width\":740,\"height\":198},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/02\\\/15\\\/tangentes-a-un-cercle-et-droites-concourantes\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Tangentes \u00e0 un cercle et droites concourantes\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\",\"name\":\"Mathweb.fr\",\"description\":\"Math\u00e9matiques, LaTeX et Python\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\",\"name\":\"St\u00e9phane Pasquet\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"width\":74,\"height\":77,\"caption\":\"St\u00e9phane Pasquet\"},\"logo\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr","description":"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/","og_locale":"fr_FR","og_type":"article","og_title":"Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr","og_description":"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.","og_url":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/","og_site_name":"Mathweb.fr","article_published_time":"2023-02-15T14:54:36+00:00","article_modified_time":"2023-02-20T13:54:31+00:00","og_image":[{"width":740,"height":198,"url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg","type":"image\/jpeg"}],"author":"St\u00e9phane Pasquet","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"St\u00e9phane Pasquet","Dur\u00e9e de lecture estim\u00e9e":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#article","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/"},"author":{"name":"St\u00e9phane Pasquet","@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"headline":"Tangentes \u00e0 un cercle et droites concourantes","datePublished":"2023-02-15T14:54:36+00:00","dateModified":"2023-02-20T13:54:31+00:00","mainEntityOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/"},"wordCount":771,"commentCount":0,"publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg","keywords":["barycentres","cercle","Ceva (th\u00e9or\u00e8me)","droites concourantes"],"articleSection":["Math\u00e9matiques"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/","url":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/","name":"Tangentes \u00e0 un cercle et droites concourantes - Mathweb.fr","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#primaryimage"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg","datePublished":"2023-02-15T14:54:36+00:00","dateModified":"2023-02-20T13:54:31+00:00","description":"Un exercice o\u00f9 il est question de tangentes \u00e0 un cercle et de droites concourantes dont la solution fait appel au th\u00e9or\u00e8me de Ceva peu connu.","breadcrumb":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#primaryimage","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/02\/head-articles-maths-tangentes-cercles-droites-concourantes.jpg","width":740,"height":198},{"@type":"BreadcrumbList","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/02\/15\/tangentes-a-un-cercle-et-droites-concourantes\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.mathweb.fr\/euclide\/"},{"@type":"ListItem","position":2,"name":"Tangentes \u00e0 un cercle et droites concourantes"}]},{"@type":"WebSite","@id":"https:\/\/www.mathweb.fr\/euclide\/#website","url":"https:\/\/www.mathweb.fr\/euclide\/","name":"Mathweb.fr","description":"Math\u00e9matiques, LaTeX et Python","publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.mathweb.fr\/euclide\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":["Person","Organization"],"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69","name":"St\u00e9phane Pasquet","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","width":74,"height":77,"caption":"St\u00e9phane Pasquet"},"logo":{"@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp"}}]}},"_links":{"self":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/8177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/comments?post=8177"}],"version-history":[{"count":0,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/8177\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media\/8182"}],"wp:attachment":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media?parent=8177"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/categories?post=8177"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/tags?post=8177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}