{"id":5277,"date":"2020-12-23T15:22:52","date_gmt":"2020-12-23T14:22:52","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=5277"},"modified":"2020-12-23T15:22:53","modified_gmt":"2020-12-23T14:22:53","slug":"la-notion-de-barycentre","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/12\/23\/la-notion-de-barycentre\/","title":{"rendered":"La notion de barycentre"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Dans la cat\u00e9gorie \u00ab\u00a0notions qui n&rsquo;est plus enseign\u00e9e en math\u00e9matiques au lyc\u00e9e\u00a0\u00bb (tiens&#8230; \u00e7a aurait \u00e9t\u00e9 une bonne id\u00e9e de cr\u00e9er cette cat\u00e9gorie sur ce site!), celle de <em>barycentre<\/em> est \u00e0 mes yeux d&rsquo;une rare beaut\u00e9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons voir dans cet article les grandes lignes ainsi que les raisons pour lesquelles cette notion est importante dans certaines sciences.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_89 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\/2020\/12\/23\/la-notion-de-barycentre\/#La_notion_de_barycentre_introduction_mathematique\" >La notion de barycentre: introduction math\u00e9matique<\/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\/2020\/12\/23\/la-notion-de-barycentre\/#Systeme_pondere\" >Syst\u00e8me pond\u00e9r\u00e9<\/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\/2020\/12\/23\/la-notion-de-barycentre\/#Barycentre_dun_systeme_pondere\" >Barycentre d&rsquo;un syst\u00e8me pond\u00e9r\u00e9<\/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\/2020\/12\/23\/la-notion-de-barycentre\/#Le_barycentre_en_astronomie\" >Le barycentre en astronomie<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/12\/23\/la-notion-de-barycentre\/#Le_barycentre_en_Physique\" >Le barycentre en Physique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/12\/23\/la-notion-de-barycentre\/#Une_propriete_mathematique\" >Une propri\u00e9t\u00e9 math\u00e9matique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/12\/23\/la-notion-de-barycentre\/#Le_systeme_de_notation_par_pouces\" >Le syst\u00e8me de notation par pouces<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_notion_de_barycentre_introduction_mathematique\"><\/span>La notion de barycentre: introduction math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Systeme_pondere\"><\/span>Syst\u00e8me pond\u00e9r\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">En math\u00e9matiques, on appelle <em>syst\u00e8me pond\u00e9r\u00e9<\/em> tout ensemble d&rsquo;\u00e9l\u00e9ments affect\u00e9 d&rsquo;un nombre, que l&rsquo;on appelle <em>poids<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;exemple le plus simple est un syst\u00e8me compos\u00e9 de notes de devoirs, chaque note \u00e9tant affect\u00e9 d&rsquo;un coefficient en fonction de l&rsquo;importance du devoir.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si Choupinette a obtenu 15\/20 (coef. 3), 13\/20 (coef. 2) et 18\/20 (coef. 1) en math\u00e9matiques au cours d&rsquo;un trimestre, on peut consid\u00e9rer que ces trois notes constitue le syst\u00e8me pond\u00e9r\u00e9:$$\\{(15,3)~;~ (13,2)~;~(18,1)\\}.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Barycentre_dun_systeme_pondere\"><\/span>Barycentre d&rsquo;un syst\u00e8me pond\u00e9r\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Le barycentre d&rsquo;un syst\u00e8me pond\u00e9r\u00e9 peut \u00eatre consid\u00e9r\u00e9 comme une <em>moyenne<\/em> du syst\u00e8me (en fait, une moyenne arithm\u00e9tique est un barycentre).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si on consid\u00e8re le syst\u00e8me pond\u00e9r\u00e9 \\(\\{(x_k,\\alpha_k)\\}_{1\\leqslant k\\leqslant n}\\), son barycentre est:$$\\text{bar}\\{(x_k,\\alpha_k)\\}_{1\\leqslant k\\leqslant n}=\\frac{1}{\\displaystyle\\sum_{k=1}^n\\alpha_k}\\sum_{k=1}^n \\alpha_kx_k.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pardon ? Que me dis-tu l\u00e0 ? Et oui&#8230; \u00e7a ressemble fortement \u00e0 une formule de statistique non ? Ben oui ! C&rsquo;est la formule qui donne la moyenne pond\u00e9r\u00e9e (je vous avais pr\u00e9venu !).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En fait, la notion de barycentre est plus li\u00e9 \u00e0 la g\u00e9om\u00e9trie qu&rsquo;aux calculs, mais l&rsquo;id\u00e9e est la m\u00eame: \u00e9tant donn\u00e9 un syst\u00e8me pond\u00e9r\u00e9 de points \\( \\{ (A_k,\\alpha_k) \\}_{1\\leqslant k\\leqslant n} \\), on d\u00e9finit le <em>barycentre<\/em> des <em>n<\/em> points comme \u00e9tant le point G tel que:$$\\sum_{k=1}^n \\alpha_k\\overrightarrow{GA_k}=\\overrightarrow{0}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_barycentre_en_astronomie\"><\/span>Le barycentre en astronomie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Notre syst\u00e8me solaire, tout comme n&rsquo;importe quel syst\u00e8me physique, a un barycentre. Tous les corps c\u00e9lestes ont une masse et donc constituent un syst\u00e8me pond\u00e9r\u00e9, qui admet un barycentre.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Everything in the Solar System orbits the center of mass (it&#039;s rarely in the center of the Sun!)\" width=\"750\" height=\"422\" src=\"https:\/\/www.youtube.com\/embed\/oauf6W3Uz04?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><figcaption>Le barycentre du syst\u00e8me solaire<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Si l&rsquo;on ne consid\u00e8re que le syst\u00e8me compos\u00e9e des plan\u00e8tes et de notre \u00e9toile (le Soleil) (les autres corps \u00e9tant de masse n\u00e9gligeable par rapport \u00e0 celle des \u00e9l\u00e9ments de ce syst\u00e8me), tous ces gravitent, tournent, autour d&rsquo;un point: le barycentre (repr\u00e9sent\u00e9 en vert dans la vid\u00e9o ci-dessus).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce constat est int\u00e9ressant \u00e0 deux points de vue:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>d&rsquo;abord, le Soleil n&rsquo;est pas le barycentre de notre syst\u00e8me, donc la prochaine fois que vous voulez briller en soci\u00e9t\u00e9, ne dites pas que toutes les plan\u00e8tes tournent autour du Soleil mais qu&rsquo;elle tournent autour d&rsquo;un point qui est tr\u00e8s proche du Soleil (en fait, il est quasi \u00e0 sa surface);<\/li><li>ensuite, si l&rsquo;on observe une \u00e9toile \u00e0 l&rsquo;ext\u00e9rieur de notre syst\u00e8me solaire, et que celle-ci semble tourner l\u00e9g\u00e8rement autour d&rsquo;un point, c&rsquo;est que cette \u00e9toile est influenc\u00e9e par d&rsquo;autres corps c\u00e9lestes de masse cons\u00e9quente (donc l&rsquo;observation d&rsquo;un barycentre d&rsquo;une \u00e9toile situ\u00e9 \u00e0 l&rsquo;ext\u00e9rieur de cette \u00e9toile montre l&rsquo;existence \u00e9ventuelle d&rsquo;autres plan\u00e8tes).<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Le constat est le m\u00eame si l&rsquo;on ne consid\u00e8re qu&rsquo;un syst\u00e8me constitu\u00e9 d&rsquo;une plan\u00e8te et de leurs lunes (comme le syst\u00e8me Terre-Lune):<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"The Earth-Moon Barycenter: The Moon doesn&#039;t orbit Earth&#039;s center and Earth doesn&#039;t stay still (3D!)\" width=\"750\" height=\"422\" src=\"https:\/\/www.youtube.com\/embed\/7hMfCCqSdFc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><figcaption>Le barycentre du syst\u00e8me Terre &#8211; Lune<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_barycentre_en_Physique\"><\/span>Le barycentre en Physique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">La formule math\u00e9matique que nous avons vue (avec des symboles de sommation discr\u00e8te \u00ab\u00a0sigma\u00a0\u00bb) peut \u00eatre envisag\u00e9e dans le cas continue (si le nombre de points est infini).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si le syst\u00e8me pond\u00e9r\u00e9 devient un ensemble continu <em>D<\/em>, constitu\u00e9 de points <em>M<\/em> \u00e0 densit\u00e9 <em>g<\/em>(<em>M<\/em>), <em>g<\/em> \u00e9tant une fonction continue (<em>champ scalaire<\/em>) alors la formule devient:$$\\int_D g(M)\\overrightarrow{GM}\\text{d}S=\\overrightarrow{0}$$ pour ce qui est dans le plan, et:$$\\int_D g(M)\\overrightarrow{GM}\\text{d}V=\\overrightarrow{0}$$pour ce qui est de l&rsquo;espace. Concr\u00e8tement, dans l&rsquo;espace \u00e0 3 dimensions (puisque c&rsquo;est l\u00e0 que nous vivons), si <em>M<\/em> a pour coordonn\u00e9es \\((x_1;x_2;x_3)\\) alors les coordonn\u00e9es dy barycentre sont donn\u00e9es par les \u00e9galit\u00e9s:$$x_{{j,{\\mathrm {G}}}}={\\frac {\\iiint g(x_{1},x_{2},x_{3})\\cdot x_{j}~{\\mathrm d}x_{1}{\\mathrm d}x_{2}{\\mathrm d}x_{3}}{\\iiint g(x_{1},x_{2},x_{3})~{\\mathrm d}x_{1}{\\mathrm d}x_{2}{\\mathrm d}x_{3}}},\\quad j\\in \\{1,2,3\\}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_propriete_mathematique\"><\/span>Une propri\u00e9t\u00e9 math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Soit le syst\u00e8me pond\u00e9r\u00e9 \\(\\{(A_k,\\alpha_k)\\}_{1\\leqslant k\\leqslant n}\\) de barycentre <em>G<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Supposons ce syst\u00e8me constitue une partition de deux ensembles disjoints (par exemple, les ensembles \\( \\{(A_1,\\alpha_1);(A_2,\\alpha_2);\\cdots;(A_{n\/2},\\alpha_{n\/2} )\\} \\) et \\( \\{(A_{n\/2+1},\\alpha_{n\/2+1});(A_{n\/2+2},\\alpha_{n\/2+2});\\cdots;(A_n,\\alpha_n) \\} \\) pour <em>n<\/em> pair, de barycentres respectifs \\(G_1\\) et \\(G_2\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Alors, $$G=\\text{bar} \\left\\{\\left(G_1;\\displaystyle\\sum_{k=1}^{n\/2}\\alpha_k\\right)~;~\\left(G_2,\\displaystyle\\sum_{k=n\/2+1}^n\\alpha_k \\right)\\right\\}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est  ce que l&rsquo;on appelle l&rsquo;associativit\u00e9 des barycentres et cela nous permet de construire plus simplement le barycentre d&rsquo;un syst\u00e8me constitu\u00e9 de 4 points par exemple:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"197\" height=\"194\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/barycentre-associativite.jpg\" alt=\"barycentre d'un syst\u00e8me \u00e0 4 points\" class=\"wp-image-5290\"\/><figcaption>Construction du barycentre d&rsquo;un syst\u00e8me \u00e0 4 points<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Explications :<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Pour construire \\(G_1=\\text{bar}\\{(A,2)~;~(B,4)\\}\\), on utilise la relation vectorielle:$$2\\overrightarrow{G_1A}+4\\overrightarrow{G_1B}=\\overrightarrow{0}$$ soit, en utilisant la relation de Chasles:$$2\\overrightarrow{G_1A}+4\\overrightarrow{G_1A}+4\\overrightarrow{AB}=\\overrightarrow{0}$$ce qui donne:$$6\\overrightarrow{G1A}=4\\overrightarrow{BA}$$et donc:$$\\overrightarrow{AG_1}=\\frac{2}{3}\\overrightarrow{AB}.$$<\/li><li>Pour construire \\(G_2=\\text{bar}\\{(C,3)~;~(D,5)\\}\\), on proc\u00e8de de m\u00eame \u00e0 partir de l&rsquo;\u00e9galit\u00e9:$$3\\overrightarrow{G_2C}+5\\overrightarrow{G_2D}=\\overrightarrow{0}$$qui donne:$$\\overrightarrow{CG_2}=\\frac{5}{8}\\overrightarrow{CD}.$$<\/li><li>Pour construire \\(G=\\text{bar}\\{(G_1,6)~;~(G_2,8)\\}\\), on proc\u00e8de de m\u00eame \u00e0 partir de l&rsquo;\u00e9galit\u00e9:$$6\\overrightarrow{GG_1}+8\\overrightarrow{GG_2}=\\overrightarrow{0}$$qui donne:$$\\overrightarrow{G_1G}=\\frac{4}{7}\\overrightarrow{G_1G_2}.$$<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_systeme_de_notation_par_pouces\"><\/span>Le syst\u00e8me de notation par pouces<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Ce syst\u00e8me repose sur la notion de barycentre. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Prenons l&rsquo;exemple d&rsquo;une plateforme de vid\u00e9os bien c\u00e9l\u00e8bre (tou doum !). Je ne suis pas dans les confidences de celle-ci mais je vais imaginer qu&rsquo;\u00e0 chaque \u0153uvre correspond un dictionnaire d&rsquo;\u00e9l\u00e9ments (genre, r\u00e9alisateur, acteurs, pays d&rsquo;origine,&#8230;). Imaginons maintenant qu&rsquo;\u00e0 chaque fois que l&rsquo;on met un pouce en l&rsquo;air pour signifier que l&rsquo;on a aim\u00e9, le compteur de chaque \u00e9l\u00e9ment soit incr\u00e9ment\u00e9. Au contraire, si on met un pouce vers le bas, les compteurs d\u00e9cr\u00e9mentent.  <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si on souhaite savoir si une autre \u0153uvre a des chances de nous plaire, on peut analyser le syst\u00e8me pond\u00e9r\u00e9 form\u00e9 par ses \u00e9l\u00e9ments (en fait, l\u00e0, concr\u00e8tement, je en vois pas trop comment mais bon&#8230; Et vous ? N&rsquo;h\u00e9sitez pas \u00e0 laisser en commentaire vos suggestions).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Document sur la notion de barycentre: <a  href=\"https:\/\/www.sbpm.be\/wp-content\/uploads\/2017\/08\/Brochure-barycentres.pdf\" target=\"_blank\" rel=\"noreferrer noopener\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">https:\/\/www.sbpm.be\/wp-content\/uploads\/2017\/08\/Brochure-barycentres.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dans la cat\u00e9gorie \u00ab\u00a0notions qui n&rsquo;est plus enseign\u00e9e en math\u00e9matiques au lyc\u00e9e\u00a0\u00bb (tiens&#8230; \u00e7a aurait \u00e9t\u00e9 une bonne id\u00e9e de cr\u00e9er cette cat\u00e9gorie sur ce site!), celle de barycentre est \u00e0 mes yeux d&rsquo;une rare beaut\u00e9. Nous allons voir dans cet article les grandes lignes ainsi que les raisons pour [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":5293,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[255,256],"class_list":["post-5277","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-astronomie","tag-physique"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Barycentre (notion) - Mathweb.fr - Astronomie, statistiques,...<\/title>\n<meta name=\"description\" content=\"La notion de barycentre sert en Astronomie, en statistiques, peut-\u00eatre m\u00eame dans le syst\u00e8me de notation par pouces en l&#039;air...\" \/>\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\/2020\/12\/23\/la-notion-de-barycentre\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Barycentre (notion) - 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