{"id":571,"date":"2018-09-02T15:42:51","date_gmt":"2018-09-02T13:42:51","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=571"},"modified":"2021-10-26T17:17:20","modified_gmt":"2021-10-26T15:17:20","slug":"le-theoreme-de-pick","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/02\/le-theoreme-de-pick\/","title":{"rendered":"Le th\u00e9or\u00e8me de Pick"},"content":{"rendered":"\n<p>On consid\u00e8re un polygone convexe, c&#8217;est-\u00e0-dire une figure g\u00e9om\u00e9trique constitu\u00e9e de plusieurs c\u00f4t\u00e9s rectilignes de sorte qu&#8217;aucun sommet ne &#8220;rentre&#8221;&nbsp; dans la figure, <strong>sur un maillage r\u00e9gulier<\/strong> de sorte que chaque sommet soit sur un n\u0153ud de&nbsp;ce maillage comme l&#8217;illustre le sch\u00e9ma ci-dessous.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01-300x300.png\" alt=\"\" class=\"wp-image-572\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01-150x150.png 150w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-01.png 321w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p>Le th\u00e9or\u00e8me de Pick stipule que <em>la superficie du polygone peut \u00eatre calcul\u00e9e de fa\u00e7on simple \u00e0 l&#8217;aide de la formule<\/em> <em>:&nbsp;&nbsp;<\/em>\\[&nbsp;\\mathcal{A}=i+\\frac{b}{2}-1\\]<br><em>exprim\u00e9e en unit\u00e9s d&#8217;aire, o\u00f9 &#8220;i&#8221; repr\u00e9sente le nombre de n\u0153uds&nbsp;int\u00e9rieurs au&nbsp;<\/em><em>polygone et &#8220;b&#8221; celui des n\u0153uds&nbsp;se trouvant sur ses c\u00f4t\u00e9s<\/em>.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>Par exemple, sur le sch\u00e9ma ci-dessus, <em>i&nbsp;<\/em>= 41 <em>b&nbsp;<\/em>= 4 ; d&#8217;o\u00f9 :&nbsp;\\[&nbsp;\\mathcal{A} = 41+2-1=42.&nbsp;\\]<br>On peut ais\u00e9ment calculer \\(\\mathcal{A}\\) en calculant d&#8217;une part l&#8217;aire du carr\u00e9 bleu ABCD (voir ci-dessous) et d&#8217;autre part, l&#8217;aire des triangles AMD, BMN, BNP, et PCD :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-02.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"276\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-02-276x300.png\" alt=\"\" class=\"wp-image-573\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-02-276x300.png 276w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-02-300x326.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-02.png 321w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><\/a><\/figure><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>Aire de ABCD : \\(8^2=64\\);<\/li><li>Aire de AMD : \\(\\frac{1}{2}\\times1\\times8=4\\);<\/li><li>Aire de BMN : \\(\\frac{1}{2}\\times7\\times2=7\\);<\/li><li>Aire de BNP : \\(\\frac{1}{2}\\times7\\times2=7\\);<\/li><li>Aire de PCD : \\(\\frac{1}{2}\\times1\\times8=4\\).<\/li><\/ul>\n\n\n\n<p>Donc l&#8217;aire de MDPN est :&nbsp;\\[&nbsp;\\mathcal{A}=64-(4+7+7+4)=64-22=42.&nbsp;\\]<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 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-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/02\/le-theoreme-de-pick\/#Demonstration\" >D\u00e9monstration<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/02\/le-theoreme-de-pick\/#La_formule_est-elle_vraie_pour_un_rectangle_quelconque\" >La formule est-elle vraie pour un rectangle quelconque ?<\/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\/2018\/09\/02\/le-theoreme-de-pick\/#La_formule_est-elle_vraie_pour_un_triangle_rectangle\" >La formule est-elle vraie pour un triangle rectangle ?<\/a><\/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\/2018\/09\/02\/le-theoreme-de-pick\/#La_formule_est-elle_vraie_pour_un_triangle_quelconque\" >La formule est-elle vraie pour un triangle quelconque ?<\/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\/2018\/09\/02\/le-theoreme-de-pick\/#Et_maintenant_le_polygone_convexe\" >Et maintenant, le polygone convexe !<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demonstration\"><\/span>D\u00e9monstration<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_formule_est-elle_vraie_pour_un_rectangle_quelconque\"><\/span>La formule est-elle vraie pour un rectangle quelconque ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Consid\u00e9rons un rectangle comme ci-dessous :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-03.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"137\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-03-300x137.png\" alt=\"\" class=\"wp-image-574\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-03-300x137.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-03-600x274.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-03.png 699w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p><br>Ici, \\(b=2(\\text{L}+1)+2(\\ell-1)=2\\text{L}+2\\ell\\) et \\(i=(\\ell-1)(\\text{L}-1)=\\text{L}\\ell-\\text{L}-\\ell+1\\).<\/p>\n\n\n\n<p>Ainsi, \\[&nbsp;\\begin{align*}i+\\dfrac{b}{2}-1 &amp; = \\text{L}\\ell-\\text{L}-\\ell+1+\\frac{2(\\text{L}+\\ell)}{2}-1\\\\&amp; = \\text{L}\\ell\\\\&amp; = \\text{aire du rectangle.}<br>\\end{align*} \\]<br>La formule de Pick est donc vraie pour un rectangle quelconque.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_formule_est-elle_vraie_pour_un_triangle_rectangle\"><\/span>La formule est-elle vraie pour un triangle rectangle ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Consid\u00e9rons la moiti\u00e9 du rectangle pr\u00e9c\u00e9dent :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-04.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"136\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-04-300x136.png\" alt=\"\" class=\"wp-image-575\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-04-300x136.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-04-600x272.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-04.png 704w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p><br>Appelons toujours&nbsp;<em>i<\/em> le nombre de points (n\u0153uds) int\u00e9rieurs et <em>b&nbsp;<\/em>= <em>c&nbsp;<\/em>+&nbsp;<em>d<\/em>&nbsp;le nombre de points sur le p\u00e9rim\u00e8tre (<em>c<\/em>&nbsp;repr\u00e9sentant le nombre de points sur les c\u00f4t\u00e9s perpendiculaires et <em>d<\/em>&nbsp;sur l&#8217;hypot\u00e9nuse).<\/p>\n\n\n\n<p>Collons \u00e0 notre triangle un autre triangle identique (bleu) pour former le rectangle initial. Appelons <em>i&#8217;<\/em>&nbsp;et <em>b&#8217;<\/em>&nbsp;les nombres respectifs de points int\u00e9rieurs au rectangle et sur le p\u00e9rim\u00e8tre.<\/p>\n\n\n\n<p>On a alors :&nbsp;\\[&nbsp;\\begin{cases}&nbsp;i&#8217;=2i+d &amp; \\text{deux fois le nombre de points int\u00e9rieurs au triangle,}\\\\&nbsp;&amp; \\text{plus ceux sur l&#8217;hypot\u00e9nuse.}\\\\&nbsp;b&#8217;=2c-2 &amp; \\text{on enl\u00e8ve deux sommets qui sont en commun.}&nbsp;\\end{cases} \\]<\/p>\n\n\n\n<p>Or, nous savons que l&#8217;aire du triangle est \u00e9gale \u00e0 la moiti\u00e9 de celle du rectangle, qui est \u00e9gale (d&#8217;apr\u00e8s la section pr\u00e9c\u00e9dente) \u00e0 \\(i&#8217;+\\frac{b&#8217;}{2}-1\\).<\/p>\n\n\n\n<p>Donc, l&#8217;aire du triangle rectangle est : \\[&nbsp;\\begin{align*}\\mathcal{A} &amp; = \\frac{1}{2}\\left(i&#8217;+\\frac{b&#8217;}{2}-1\\right)\\\\&nbsp;&amp; = \\frac{1}{2}\\left(2i+d+\\frac{2c-2}{2}-1\\right)\\\\&nbsp;&amp; = \\frac{1}{2}\\left(2i+d+c-1-1\\right)\\\\&nbsp;&amp; = i+\\frac{b}{2}-1\\text{ car }d+c=b.&nbsp;\\end{align*} \\]<\/p>\n\n\n\n<p>Le th\u00e9or\u00e8me de Pick est donc vrai pour le triangle rectangle.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_formule_est-elle_vraie_pour_un_triangle_quelconque\"><\/span>La formule est-elle vraie pour un triangle quelconque ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Consid\u00e9rons maintenant un rectangle quelconque comme celui repr\u00e9sent\u00e9 ci-dessous :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-05.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"108\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-05-300x108.png\" alt=\"\" class=\"wp-image-576\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-05-300x108.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-05-600x216.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-05.png 721w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p><br>L&#8217;aire du triangle BMP est la diff\u00e9rence entre celle du rectangle ABCD et de la somme de celle des triangles rectangles AMB (triangle 1), BCP (triangle 2) et PMD (triangle 3).<\/p>\n\n\n\n<p>Appelons :<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\\(i_k\\) le nombre de points \u00e0 l&#8217;int\u00e9rieur du triangle <em>k<\/em>&nbsp;et \\(b_k = c_k+d_k\\) le nombre de points sur les bords, avec \\(d_k\\) repr\u00e9sentant le nombre de points sur son hypot\u00e9nuse (extr\u00e9mit\u00e9s non comprises) et \\(c_k\\) repr\u00e9sentant le nombre de points sur ces autres c\u00f4t\u00e9s, <em>k<\/em>&nbsp;\u00e9tant un entier \u00e9gal \u00e0 1, 2 ou 3 ;<\/li><li><em>i<\/em>&nbsp;le nombre de points \u00e0 l&#8217;int\u00e9rieur du triangle quelconque et \\(b=d_1+d_2+d_3+3\\) le nombre de points sur ses bords ;<\/li><li><em>i&#8217;<\/em>&nbsp;le nombre de points \u00e0 l&#8217;int\u00e9rieur du rectangle et <em>b&#8217;<\/em>&nbsp;le nombre de points sur ses bords.<\/li><\/ul>\n\n\n\n<p>Notons \\(\\mathcal{A}\\) l&#8217;aire du triangle BMP, \\(\\mathcal{A}_k\\) celle du triangle&nbsp;<em>k<\/em> et \\(\\mathcal{A}^\\prime\\) celle du rectangle ABCD. Alors : \\[&nbsp;\\begin{align*}&nbsp;\\mathcal{A} &amp; = \\mathcal{A}^\\prime &#8211; (\\mathcal{A}_1+\\mathcal{A}_2+\\mathcal{A}_3)\\\\&nbsp;&amp; = \\left(i&#8217;+\\frac{b&#8217;}{2}-1\\right)-\\left(i_1+i_2+i_3+\\frac{1}{2}(b_1+b_2+b_3)-3\\right)\\\\&nbsp;&amp; = i&#8217;+\\frac{b&#8217;}{2}-1-\\left(i_1+i_2+i_3+\\frac{1}{2}(c_1+c_2+c_3+d_1+d_2+d_3) -3\\right).\\end{align*}\\]<\/p>\n\n\n\n<p>Or,&nbsp;\\[ i&#8217;=i+i_1+i_2+i_3+d_1+d_2+d_3 \\]&nbsp;et&nbsp;\\[ b&#8217;=c_1+c_2+c_3-3, \\]<br>donc : \\[&nbsp;\\begin{align*}&nbsp;\\mathcal{A} &amp; = i+i_1+i_2+i_3+d_1+d_2+d_3+\\frac{c_1+c_2+c_3-3}{2}-1-i_1-i_2-i_3-\\frac{c_1+c_2+c_3+d_1+d_2+d_3}{2}+3\\\\&nbsp;&amp; = i+\\frac{d_1+d_2+d_3+3}{2}-1\\\\&nbsp;&amp; = i+\\frac{b}{2}-1.&nbsp;\\end{align*} \\]<\/p>\n\n\n\n<p>Nous avons ainsi d\u00e9montr\u00e9 que le th\u00e9or\u00e8me de Pick \u00e9tait vrai pour un triangle quelconque.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Et_maintenant_le_polygone_convexe\"><\/span>Et maintenant, le polygone convexe !<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Un polygone convexe P \u00e0&nbsp;<em>n<\/em> c\u00f4t\u00e9s est form\u00e9 de <em>n<\/em>-2 triangles, <em>n &gt;&nbsp;<\/em>3. Notons \\(T_k\\) les <em>n<\/em>-2 triangles qui constituent le polygone. Je vais ici prendre un hexagone pour avoir un appui visuel :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-06.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"274\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-06-300x274.png\" alt=\"\" class=\"wp-image-577\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-06-300x274.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/theoreme-de-pick-06.png 352w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p><br>Je conviens alors de d\u00e9couper&nbsp;<em>P<\/em> \u00e0 partir d&#8217;un m\u00eame point A et en nommant les triangles de gauche \u00e0 droite \\(T_1,\\ T_2,\\ \\ldots,\\ T_{n-2}\\).<\/p>\n\n\n\n<p>Je note alors :<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\\(c_1\\) le nombre de points sur les deux c\u00f4t\u00e9s de \\(T_1\\) qui co\u00efncident avec deux c\u00f4t\u00e9s de <em>P<\/em>&nbsp;;<\/li><li>\\(c_{n-2}\\) le nombre de points sur les deux c\u00f4t\u00e9s de \\(T_{n-2}\\) qui co\u00efncident avec deux c\u00f4t\u00e9s de <em>P<\/em>&nbsp;;<\/li><li>\\(c_k\\), \\(1\\leq k \\leq n-3\\), le nombre de points sur le seul c\u00f4t\u00e9 de \\(T_k\\) qui co\u00efncide avec un c\u00f4t\u00e9 de <em>P<\/em>&nbsp;;<\/li><li>\\(d_k\\), \\(1\\leq k \\leq n-3\\), le nombre de points sur le c\u00f4t\u00e9 de \\(T_k\\) int\u00e9rieur \u00e0 <em>P<\/em>&nbsp;qui ne co\u00efncide pas avec un c\u00f4t\u00e9 de \\(T_{k-1}\\) ;<\/li><li>\\(i_k\\) le nombre de points int\u00e9rieurs \u00e0 \\(T_k\\) ;<\/li><li><em>i<\/em>&nbsp;le nombre de points int\u00e9rieurs \u00e0&nbsp;<em>P<\/em> ;<\/li><li><em>b<\/em>&nbsp;le nombre de points sur les bords de <em>P<\/em>.<\/li><\/ul>\n\n\n\n<p>On a alors :&nbsp;\\[&nbsp;i=\\sum_{k=1}^{n-2} i_k+\\sum_{k=1}^{n-3} d_k&nbsp;\\]&nbsp; et&nbsp;\\[ b=c_1+(c_2-1)+(c_3-1)+\\cdots+(c_{n-3}-1)+(c_{n-2}-2)\\;,&nbsp;\\]<br>soit :&nbsp;\\[ b=\\sum_{k=1}^{n-2} c_k -(n-2)\\;,\\]<br>ou encore :&nbsp;\\[ b=\\sum_{k=1}^{n-2}c_k-n+2.\\]<br>En notant \\(\\mathcal{A}\\) l&#8217;aire de&nbsp;<em>P<\/em> et \\(\\mathcal{A}_k\\) celle de \\(T_k\\), on a : \\[&nbsp;\\begin{align*}\\mathcal{A} &amp; = \\sum_{k=1}^{n-2}\\mathcal{A}_k\\\\&nbsp;&amp; = \\mathcal{A}_1+ \\sum_{k=2}^{n-3}\\mathcal{A}_k+\\mathcal{A}_{n-2}\\\\&nbsp;&amp; = i_1+\\frac{b_1}{2}-1+\\sum_{k=2}^{n-3}\\mathcal{A}_k+i_{n-2}+\\frac{b_{n-2}}{2}-1\\\\&nbsp;&amp; = i_1+i_{n-2}+\\frac{b_1+b_{n-2}}{2}-2+\\sum_{k=2}^{n-3}\\left(i_k+\\frac{b_k}{2}-1\\right)\\\\<br>&amp; = i_1+i_{n-2}+\\frac{b_1+b_{n-2}}{2}-2+\\sum_{k=2}^{n-3}\\left( i_k+\\frac{d_{k-1}+c_k+d_k+1}{2}-1\\right)\\\\&nbsp;&amp; = i_1+i_{n-2}+\\frac{b_1+b_{n-2}}{2}-2+\\sum_{k=2}^{n-3}i_k + \\frac{1}{2}\\sum_{k=2}^{n-3}c_k + \\frac{1}{2}\\sum_{k=2}^{n-3}(d_{k-1}+d_k)\\\\&amp;\\qquad -\\frac{1}{2}(n-4) \\\\&nbsp;&amp; = \\sum_{k=1}^{n-2}i_k+\\frac{c_1+d_1+c_{n-2}+d_{n-3}}{2}+\\frac{1}{2}\\sum_{k=2}^{n-3}c_k+\\frac{1}{2}\\sum_{k=2}^{n-3}(d_{k-1}+d_k)\\\\&amp;\\qquad-\\frac{1}{2}n\\\\&nbsp;&amp; = \\sum_{k=1}^{n-2}i_k+\\frac{1}{2}\\sum_{k=1}^{n-2}c_k + \\sum_{k=1}^{n-2}d_k-\\frac{1}{2}n\\\\&nbsp;&amp; = \\left(\\sum_{k=1}^{n-2}i_k+\\sum_{k=1}^{n-3}d_k\\right)+\\frac{1}{2}\\sum_{k=1}^{n-2}c_k-\\frac{1}{2}n\\\\&nbsp;&amp; = i+\\frac{b+n-2}{2}-\\frac{1}{2}n\\\\<br>&amp; = i+\\frac{b}{2}-1.&nbsp;\\end{align*} \\]<br>Le th\u00e9or\u00e8me de Pick est donc d\u00e9montr\u00e9 pour tout polygone convexe.<\/p>\n\n\n\n<p>T\u00e9l\u00e9charger cet article au format PDF :&nbsp;<a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Th\u00e9or\u00e8me-de-Pick.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Th\u00e9or\u00e8me de Pick<\/a><\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Th\u00e9or\u00e8me-de-Pick.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Th\u00e9or\u00e8me de Pick<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Th\u00e9or\u00e8me-de-Pick.pdf\" class=\"wp-block-file__button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p>Obtenir les sources \\(\\LaTeX\\) du document PDF :<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\" id=\"um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Th\u00e9or\u00e8me-de-Pick.zip\">Th\u00e9or\u00e8me de Pick<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Th\u00e9or\u00e8me-de-Pick.zip\" class=\"wp-block-file__button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>On consid\u00e8re un polygone convexe, c&#8217;est-\u00e0-dire une figure g\u00e9om\u00e9trique constitu\u00e9e de plusieurs c\u00f4t\u00e9s rectilignes de sorte qu&#8217;aucun sommet ne &#8220;rentre&#8221;&nbsp; dans la figure, sur un maillage r\u00e9gulier de sorte que chaque sommet soit sur un n\u0153ud de&nbsp;ce maillage comme l&#8217;illustre le sch\u00e9ma ci-dessous. Le th\u00e9or\u00e8me de Pick stipule que la [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[54,25,46],"class_list":["post-571","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-aire","tag-demonstration","tag-triangle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Le th\u00e9or\u00e8me de Pick - Mathweb.fr<\/title>\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\/2018\/09\/02\/le-theoreme-de-pick\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le th\u00e9or\u00e8me de Pick - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"On consid\u00e8re un polygone convexe, c&#8217;est-\u00e0-dire une figure g\u00e9om\u00e9trique constitu\u00e9e de plusieurs c\u00f4t\u00e9s rectilignes de sorte qu&#8217;aucun sommet ne &#8220;rentre&#8221;&nbsp; dans la figure, sur un maillage r\u00e9gulier de sorte que chaque sommet soit sur un n\u0153ud de&nbsp;ce maillage comme l&#8217;illustre le sch\u00e9ma ci-dessous. 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