{"id":10593,"date":"2025-10-09T19:03:28","date_gmt":"2025-10-09T17:03:28","guid":{"rendered":"https:\/\/spgoo.org\/?page_id=10593"},"modified":"2025-10-09T19:03:28","modified_gmt":"2025-10-09T17:03:28","slug":"moy-arithmetique-moy-geometrique","status":"publish","type":"page","link":"https:\/\/spgoo.org\/?page_id=10593","title":{"rendered":"Moy arithm\u00e9tique &#8211; Moy G\u00e9om\u00e9trique"},"content":{"rendered":" \r\n  <script src=\"https:\/\/cdn.jsdelivr.net\/npm\/jsxgraph\/distrib\/jsxgraphcore.js\"><\/script>\r\n  <link rel=\"stylesheet\" href=\"https:\/\/cdn.jsdelivr.net\/npm\/jsxgraph\/distrib\/jsxgraph.css\" \/>\r\n  <script src=\"https:\/\/cdn.jsdelivr.net\/npm\/mathjax@3\/es5\/tex-mml-chtml.js\"><\/script>\r\n  <style>\r\n    #jxgbox {\r\n      width: 800px;\r\n      height: 800px;\r\n      margin: auto;\r\n      border: 1px solid #ccc;\r\n    }\r\n  <\/style>\r\n \r\n  <div id=\"jxgbox\"><\/div>\r\n\r\n  <script>\r\n    JXG.Options.text.useMathJax = true;\r\n    const board = JXG.JSXGraph.initBoard('jxgbox', {\r\n      boundingbox: [-6, 5, 1.5, -0],\r\n      axis:false,\r\n      showNavigation: false,\r\n      showCopyright: false,\r\n      keepAspectRatio:true\r\n    });\r\n    board.renderer.container.style.backgroundColor = 'gainsboro';\r\n    JXG.Options.point.showInfobox = false;\r\n\r\nconst titr0 = board.create('text', [-2,6,'\\\\(\\\\textrm{In\u00e9galit\u00e9 des Moyennes Arithm\u00e9tique et G\u00e9om\u00e9triques pour } n= 2 \\\\)'], {\r\ncolor:\"#AE181E\",\r\nfontSize:18,\r\nfixed:true,\r\nanchorX:'middle'\r\n});\r\n\r\nconst p0 = board.create('point', [-5.5, 0],{visible:false});\r\nconst p1 = board.create('point', [-5.5, 5.5],{visible:false});\r\nconst p2 = board.create('point', [0, 5.5],{visible:false});\r\nconst p3 = board.create('point', [0, 0],{visible:false});\r\nconst pol1 = board.create('polygon', [p0, p1,p2,p3],{\r\n  borders: {\r\n    strokeColor: 'transparent', \/\/ rend la bordure invisible\r\n    highlightStrokeColor: 'transparent' \/\/ emp\u00eache la bordure de s'afficher au survol\r\n  },\r\n  fillColor: 'lightblue', \/\/ couleur de remplissage\r\n  highlight: false,\r\n  shadow:true\r\n});\r\n\r\nconst s0 = board.create('segment', [p0, p1]);\r\nconst s1 = board.create('segment', [p1, p2]);\r\nconst s2 = board.create('segment', [p2, p3]);\r\nconst s3 = board.create('segment', [p3, p0]);\r\n\r\nconst gl0 =  board.create('glider',[-5.5,1, s0],{\r\n      name:\"gl0\",\r\n\t  highlight:false,\r\n\t  fillColor:\"#AE181E\",\r\n\t  strokeColor:\"navy\",\r\n\t  showInfobox: false, \r\n\t  size:4,\r\n\t  visible:false\r\n\t  });\r\n  \r\n\r\n\r\nconst gl1 = board.create('glider',[0, gl0.Y(), s2],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:true\r\n});\r\n\r\nconst l0 = board.create(\"line\", [\r\n  function() { return [gl0.X(), gl0.Y()]; },\r\n  function() { return [gl1.X(), gl1.Y()]; }\r\n], {\r\n  straightFirst: false,\r\n  straightLast: false,\r\n  strokeColor: \"#333\",\r\n  strokeWidth: 2,\r\n  visible:false\r\n});\r\n\r\n\r\n\r\n\/\/ Quand gl1 est d\u00e9plac\u00e9, gl0 suit en Y\r\ngl1.on('drag', function() {\r\n  gl0.moveTo([-5.5, gl1.Y()]);\r\n});\r\n\r\nconst gl2 = board.create('glider',[-gl1.Y(), 0, s3],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:false\r\n});\r\nconst s6 =  board.create('segment',[p3, gl2],{name:\"\",highlight:false,strokeColor:\"#AE181E\",strokeWidth:4});\r\nconst s7 =  board.create('segment',[p0, gl2],{name:\"\",highlight:false,strokeColor:\"navy\",strokeWidth:4});\r\nconst gl3 = board.create('glider',[gl2.X(), 5.5, s1],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:true\r\n});\r\n\r\nconst l1 = board.create(\"line\", [\r\n  function() { return [gl2.X(), gl2.Y()]; },\r\n  function() { return [gl3.X(), gl3.Y()]; }\r\n], {\r\n  straightFirst: false,\r\n  straightLast: false,\r\n  strokeColor: \"#333\",\r\n  strokeWidth: 2,\r\n  visible:false\r\n});\r\n\r\nconst s8 =  board.create('segment',[p2, gl3],{name:\"\",highlight:false,strokeColor:\"#AE181E\",strokeWidth:4});\r\nconst s9 =  board.create('segment',[p1, gl3],{name:\"\",highlight:false,strokeColor:\"navy\",strokeWidth:4});\r\nconst s10 =  board.create('segment',[p3, gl1],{name:\"\",highlight:false,strokeColor:\"#AE181E\",strokeWidth:4});\r\nconst s11 =  board.create('segment',[p2, gl1],{name:\"\",highlight:false,strokeColor:\"navy\",strokeWidth:4});\r\n\r\nconst gl4 = board.create('glider',[-5.5+gl0.Y(), 0, s3],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:true\r\n});\r\n\r\nconst gl5 = board.create('glider',[-5.5+gl0.Y(), 5.5, s1],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:false\r\n});\r\n\r\nconst l2 = board.create(\"line\", [\r\n  function() { return [gl4.X(), gl4.Y()]; },\r\n  function() { return [gl5.X(), gl5.Y()]; }\r\n], {\r\n  straightFirst: false,\r\n  straightLast: false,\r\n  strokeColor: \"#333\",\r\n  strokeWidth: 2,\r\n  visible:false\r\n});\r\n\r\nconst gl6 = board.create('glider',[-5.5, 5.5-gl0.Y(), s0],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:true\r\n});\r\n\r\nconst gl7 = board.create('glider',[-5.5,5.5-gl0.Y(), 5.5, s2],{\r\n  name:\"\",\r\n  highlight:false,\r\n  fillColor:\"#AE181E\",\r\n  strokeColor:\"navy\",\r\n  showInfobox: false,\r\n  size:4,\r\n  snapToGrid: false,\r\n  visible:false\r\n});\r\n\r\nconst l3 = board.create(\"line\", [\r\n  function() { return [gl6.X(), gl6.Y()]; },\r\n  function() { return [gl7.X(), gl7.Y()]; }\r\n], {\r\n  straightFirst: false,\r\n  straightLast: false,\r\n  strokeColor: \"#333\",\r\n  strokeWidth: 2,\r\n  visible:false\r\n});\r\n\r\nconst inter0 = board.create('intersection',[l0,l2],{visible:false});\r\nconst inter1 = board.create('intersection',[l3,l2],{visible:false});\r\nconst inter2 = board.create('intersection',[l3,l1],{visible:false});\r\nconst inter3 = board.create('intersection',[l0,l1],{visible:false});\r\nconst pol0 = board.create('polygon',[inter0,inter1,inter2,inter3],{highlight:false,color:\"slategray\"});\r\n\r\nconst seg0 = board.create('segment',[gl6,inter1],{highlight:false,color:\"slategray\",dash:2});\r\nconst seg1 = board.create('segment',[gl3,inter2],{highlight:false,color:\"slategray\",dash:2});\r\nconst seg2 = board.create('segment',[gl1,inter3],{highlight:false,color:\"slategray\",dash:2});\r\nconst seg3 = board.create('segment',[gl4,inter0],{highlight:false,color:\"slategray\",dash:2});\r\n\r\n\r\n\/\/ D\u00e9placement des gliders\r\ngl0.on('drag', function() {\r\n  gl1.moveTo([0,gl0.Y()]);\r\n  gl2.moveTo([-gl0.Y(),0]);\r\n  gl3.moveTo([-gl0.Y(),5.5]);\r\n  gl4.moveTo([-5.5+gl0.Y(),5.5]);\r\n  gl5.moveTo([-5.5+gl0.Y(),0]);\r\n  gl6.moveTo([-5.5,5.5-gl0.Y()]);\r\n  gl7.moveTo([0,5.5-gl0.Y()]);\r\n});\r\n \r\ngl1.on('drag', function() {\r\n  gl0.moveTo([-5.5,gl1.Y()]);\r\n  gl2.moveTo([-gl1.Y(),0]);\r\n  gl3.moveTo([-gl1.Y(),5.5]);\r\n  gl4.moveTo([-5.5+gl1.Y(),5.5]);\r\n  gl5.moveTo([-5.5+gl1.Y(),0]);\r\n  gl6.moveTo([-5.5,5.5-gl1.Y()]);\r\n  gl7.moveTo([0,5.5-gl1.Y()]); \r\n});\r\n\r\ngl2.on('drag', function() {\r\n  gl0.moveTo([-5.5,-gl2.X()]);\r\n  gl1.moveTo([0,-gl2.X()]);\r\n  gl3.moveTo([gl2.X(),5.5]);\r\n  gl4.moveTo([-5.5-gl2.X(),0]);\r\n  gl5.moveTo([-5.5-gl2.X(),5.5]);\r\n  gl6.moveTo([-5.5,5.5+gl2.X()]);\r\n  gl7.moveTo([0,5.5+gl2.X()]);  \r\n});\r\n\r\ngl3.on('drag', function() {\r\n  gl0.moveTo([-5.5, -gl3.X()]);\r\n  gl1.moveTo([0,-gl3.X()]);\r\n  gl2.moveTo([gl3.X(),5.5]);\r\n  gl4.moveTo([-5.5-gl3.X(),0]);\r\n  gl5.moveTo([-5.5-gl3.X(),5.5]);\r\n  gl6.moveTo([-5.5,5.5+gl3.X()]);\r\n  gl7.moveTo([0,5.5+gl3.X()]);  \r\n});\r\n\r\ngl4.on('drag', function() {\r\n  gl0.moveTo([-5.5,5.5+gl4.X()]);\r\n  gl1.moveTo([0,5.5+gl4.X()]);\r\n  gl2.moveTo([-5.5-gl4.X(),0]);\r\n  gl3.moveTo([-5.5-gl4.X(),5.5]);\r\n  gl5.moveTo([gl4.X(),5.5]);\r\n  gl6.moveTo([-5.5,-gl4.X()]);\r\n  gl7.moveTo([0,-gl4.X()]);  \r\n});\r\n\r\n\r\ngl5.on('drag', function() {\r\n  gl0.moveTo([-5.5,5.5+gl5.X()]);\r\n\t  gl1.moveTo([0,5.5+gl5.X()]);\r\n\t  gl2.moveTo([-5.5-gl5.X(),0]);\r\n\t  gl3.moveTo([-5.5-gl5.X(),5.5]);\r\n\t  gl4.moveTo([gl5.X(),0]);\r\n\t  gl6.moveTo([-5.5,-gl5.X()]);\r\n  gl7.moveTo([0,-gl5.X()]);  \r\n});\r\n\r\ngl6.on('drag', function() {\r\n  gl0.moveTo([-5.5,5.5-gl6.Y()]);\r\n  gl1.moveTo([0,5.5-gl6.Y()]);\r\n  gl2.moveTo([-5.5+gl6.Y(),0]);\r\n  gl3.moveTo([-5.5+gl6.Y(),5.5]);\r\n  gl4.moveTo([-gl6.Y(),0]);\r\n  gl5.moveTo([-gl6.Y(),5.5]);\r\n  gl7.moveTo([0,gl6.Y()]);  \r\n});\r\n\r\ngl7.on('drag', function() {\r\n  gl0.moveTo([-5.5,5.5-gl7.Y()]);\r\n  gl1.moveTo([0,5.5-gl7.Y()]);\r\n  gl2.moveTo([-5.5+gl7.Y(),0]);\r\n  gl3.moveTo([-5.5+gl7.Y(),5.5]);\r\n  gl4.moveTo([-gl7.Y(),0]);\r\n  gl5.moveTo([-gl7.Y(),5.5]);\r\n  gl6.moveTo([0,gl7.Y()]);  \r\n});\r\n\r\nconst s4 =  board.create('segment',[p0, gl6],{name:\"\",highlight:false,strokeColor:\"#AE181E\",strokeWidth:4});\r\nconst mp0 = board.create('midpoint', [s4],{name:'',visible:false});\r\nconst t0 = board.create('text', [function(){return mp0.X()-0.3;},function(){return mp0.Y();},'\\\\(a\\\\)'],\r\n{visible: function() {\r\n    return !(gl6.X() === p0.X() && gl6.Y() === 0);\r\n  },color:\"#AE181E\",fontSize:16,fixed:true,});\r\n\r\ngl6.on('update', function() {\r\n  if (gl6.X() === p0.X() && gl6.Y() === p0.Y()) {\r\n    t0.setAttribute({visible: false});\r\n  } else {\r\n    t0.setAttribute({visible: true});\r\n}\r\nboard.update(); \r\n});  \r\n\r\nconst s5 =  board.create('segment',[gl6,p1],{name:\"\",highlight:false,strokeColor:\"navy\",strokeWidth:4});\r\nconst mp1 = board.create('midpoint', [s5],{name:'',visible:false});\r\nconst t1 = board.create('text', [function(){return mp1.X()-0.3;},function(){return mp1.Y();},'\\\\(b\\\\)'],{visible: function() {\r\n    return !(gl6.X() === p1.X() && gl6.Y() === p1.Y());\r\n  },color:\"navy\",fontSize:16,fixed:true,});\r\n  \r\ngl0.on('update', function() {\r\n  if (gl6.X() === p1.X() && gl6.Y() === p1.Y()) {\r\n    t1.setAttribute({visible: false});\r\n  } else {\r\n    t1.setAttribute({visible: true});\r\n}\r\nboard.update(); \r\n});  \r\n\r\nconst s12 =  board.create('segment',[gl1,gl7],{name:\"\",highlight:false,strokeColor:\"forestgreen\",strokeWidth:4});\r\nconst mp2 = board.create('midpoint', [s12],{name:'',visible:false});\r\nconst t2 = board.create('text', [\r\n  function() { return mp2.X() + 0.3; },\r\n  function() { return mp2.Y(); },\r\n  '\\\\(a-b\\\\)'\r\n], {color:\"forestgreen\",fontSize:16,fixed:true,\r\n  visible: function() {\r\n    return gl1.Y() <= gl7.Y(); \/\/ t2 visible si gl1 est en dessous ou \u00e9gal\r\n  }\r\n});\r\n\r\nconst t3 = board.create('text', [\r\n  function() { return mp2.X() + 0.3; },\r\n  function() { return mp2.Y(); },\r\n  '\\\\(b-a\\\\)'\r\n], {color:\"forestgreen\",fontSize:16,fixed:true,\r\n  visible: function() {\r\n    return gl1.Y() > gl7.Y(); \/\/ t3 visible si gl1 est au-dessus\r\n  }\r\n});\r\n \r\n const t4 = board.create('text', [\r\n  function() { return gl7.X() + 0.5; },\r\n  function() { return gl7.Y(); },\r\n  '\\\\(0\\\\)'\r\n], {fixed:true,\r\n  visible: function() {\r\n    return gl1.Y() === gl7.Y(); \/\/ \u2705 comparaison correcte\r\n  }\r\n});\r\n \r\nconst polyhash0 = board.create('polygon', [p0, gl6, inter1, gl4], {\r\n  fillColor: 'brown',\r\n  borders: {strokeColor: 'transparent'},\r\n  highlight: false,\r\n  opacity:0.1\r\n}); \r\n\r\nconst polyhash1 = board.create('polygon', [gl6, p1, gl3, inter2], {\r\n  fillColor: '#FFD700',\r\n  borders: {strokeColor: 'transparent'},\r\n  highlight: false,\r\n  opacity:0.1\r\n}); \r\n\r\nconst polyhash2 = board.create('polygon', [gl6, p1, gl3, inter2], {\r\n  fillColor: '#FFD700',\r\n  borders: {strokeColor: 'transparent'},\r\n  highlight: false,\r\n  opacity:0.1\r\n}); \r\n\r\n const polyhash3 = board.create('polygon', [gl3, p2,gl1,inter3], {\r\n  fillColor: '#E30B5C',\r\n  borders: {strokeColor: 'transparent'},\r\n  highlight: false,\r\n  opacity:0.1\r\n}); \r\n\r\n const polyhash4 = board.create('polygon', [gl1, p3,gl4,inter0], {\r\n  fillColor: '#ECFFDC',\r\n  borders: {strokeColor: 'transparent'},\r\n  highlight: false,\r\n  opacity:0.1\r\n});\r\n\r\n\/*\r\nconst extex0 = board.create('text', [-2.5,-.5,\r\n  '\\\\(4 \\\\times (a\\\\times b)\\\\leq (a+b)^2 \\\\,\\\\, \\\\Longleftrightarrow \\\\,\\\\, \\\\sqrt{a\\\\times b}\\\\leq \\\\frac{a+b}{2} \\\\,\\\\, \\\\Longleftrightarrow \\\\,\\\\, \\\\textrm{Moyenne G\u00e9om\u00e9trique}\\\\,\\\\, \\\\leq \\\\,\\\\, \\\\textrm{Moyenne Arithm\u00e9tique} \\\\)'\r\n], {color:\"navy\",fontSize:14,anchorX:'middle'\r\n});\r\n\r\n*\/\r\n\r\nconst textParts = [\r\n  '\\\\(4 \\\\times (a\\\\times b)\\\\leq (a+b)^2\\\\)',\r\n  '\\\\(\\\\Longleftrightarrow\\\\,\\\\, \\\\sqrt{a\\\\times b}\\\\leq \\\\frac{a+b}{2}\\\\,\\\\,\\\\)',\r\n  '<span style=\"color:#AE181E\">\\\\(\\\\Longleftrightarrow\\\\,\\\\,\\\\)<\/span>',\r\n  '<span style=\"color:#AE181E\">\\\\(\\\\textrm{Moyenne G\u00e9om\u00e9trique}\\\\leq \\\\textrm{Moyenne Arithm\u00e9tique}\\\\)<\/span>'\r\n];\r\n\r\nlet currentText = '';\r\nconst extex0 = board.create('text', [-2.5, -0.5, ''], {\r\n  fontSize: 14,\r\n  anchorX: 'middle'\r\n});\r\n\r\n\/\/ Textes suppl\u00e9mentaires, initialement invisibles\r\nconst extex1 = board.create('text', [-5.25, -0.8, '\\\\(\\\\uparrow\\\\)'], {\r\n  color: \"navy\", fontSize: 14, anchorX: 'middle', visible: false,fixed:true\r\n});\r\nconst extex2 = board.create('text', [-5.0, -1.1, '\\\\(\\\\textrm{Surface des rectangles color\u00e9s}\\\\)'], {\r\n  color: \"navy\", fontSize: 14, anchorX: 'middle', visible: false,fixed:true\r\n});\r\nconst extex3 = board.create('text', [-4.0, -0.9, '\\\\(\\\\textrm{Surface du carr\u00e9 ext\u00e9rieur}\\\\)'], {\r\n  color: \"navy\", fontSize: 14, anchorX: 'middle', visible: false,fixed:true\r\n});\r\nconst extex4 = board.create('text', [-4.55, -0.7, '\\\\(\\\\uparrow\\\\)'], {\r\n  color: \"navy\", fontSize: 14, anchorX: 'middle', visible: false,fixed:true\r\n});\r\n\r\n\/\/ Affichage progressif du texte principal\r\nfunction showNextPart(index) {\r\n  if (index >= textParts.length) {\r\n    \/\/ Une fois le texte principal termin\u00e9, lancer les autres textes\r\n    setTimeout(() => { extex1.setAttribute({visible: true}); board.update(); }, 1000);\r\n    setTimeout(() => { extex2.setAttribute({visible: true}); board.update(); }, 2500);\r\n    setTimeout(() => { extex3.setAttribute({visible: true}); board.update(); }, 4000);\r\n    setTimeout(() => { extex4.setAttribute({visible: true}); board.update(); }, 5500);\r\n    return;\r\n  }\r\n\r\n  currentText += textParts[index] + ' ';\r\n  extex0.setText(currentText);\r\n  board.update();\r\n\r\n  setTimeout(() => {\r\n    showNextPart(index + 1);\r\n  }, 1500); \/\/ d\u00e9lai entre chaque partie\r\n}\r\n\r\n\/\/ D\u00e9marrage apr\u00e8s une courte pause\r\nsetTimeout(() => {\r\n  showNextPart(0);\r\n}, 1000);\r\n \r\n \r\n <\/script>\r\n\r\n\r\n\r\n\r\n\r\n\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-10593","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/pages\/10593","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spgoo.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10593"}],"version-history":[{"count":1,"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/pages\/10593\/revisions"}],"predecessor-version":[{"id":10594,"href":"https:\/\/spgoo.org\/index.php?rest_route=\/wp\/v2\/pages\/10593\/revisions\/10594"}],"wp:attachment":[{"href":"https:\/\/spgoo.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10593"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}