{"version":3,"file":"6G14-eZRiDtBQ.js","sources":["../../src/exercices/6e/6G14.js"],"sourcesContent":["import { codageAngleDroit } from '../../lib/2d/angles.js'\nimport {\n  droite,\n  droiteParPointEtParallele,\n  droiteParPointEtPente,\n  droiteParPointEtPerpendiculaire\n} from '../../lib/2d/droites.js'\nimport { point, pointIntersectionDD, pointSurDroite } from '../../lib/2d/points.js'\nimport { segment } from '../../lib/2d/segmentsVecteurs.js'\nimport { latexParCoordonneesBox } from '../../lib/2d/textes.js'\nimport { rotation } from '../../lib/2d/transformations.js'\nimport { choice, combinaisonListesSansChangerOrdre, shuffle } from '../../lib/outils/arrayOutils'\nimport { range, rangeMinMax } from '../../lib/outils/nombres'\nimport Exercice from '../deprecatedExercice.js'\nimport { mathalea2d, vide2d } from '../../modules/2dGeneralites.js'\nimport { context } from '../../modules/context.js'\nimport {\n  listeQuestionsToContenu,\n  randint,\n  egal,\n  contraindreValeur,\n  gestionnaireFormulaireTexte\n} from '../../modules/outils.js'\n\nexport const amcReady = true\nexport const amcType = 'AMCOpen'\nexport const titre = 'Use the properties of perpendicular lines'\n\n/**\n * Ref 6G14\n * @author Jean-Claude Lhote (EE : pour l'ajout d'AMC et la possibilité de sélectionner différents mélanges)\n * @author Mickael Guironnet (refactoring avec ajout des 4 à 6 et des figures)\n * publié le 22/11/2020\n */\nexport const uuid = '6a336'\nexport const ref = '6G14'\nexport default function ProprietesParallelesPerpendiculaires () {\n  Exercice.call(this)\n  this.nbQuestions = 3\n  this.nbCols = 1\n  this.nbColsCorr = 1\n  this.sup = 4\n  this.sup2 = false\n  this.sup3 = true\n  this.correctionDetailleeDisponible = true\n  this.correctionDetaillee = false\n  this.nouvelleVersion = function () {\n    this.autoCorrection = []\n    const typesDeQuestionsDisponibles = []\n    let questionsParNiveau = []\n    if (!this.sup2) {\n      questionsParNiveau.push(range(3))\n      questionsParNiveau.push(rangeMinMax(4, 6))\n      questionsParNiveau.push(rangeMinMax(9, 15))\n      questionsParNiveau.push(rangeMinMax(19, 31, 20))\n      questionsParNiveau.push(questionsParNiveau[0].concat(questionsParNiveau[1].concat(questionsParNiveau[2]).concat(questionsParNiveau[3])))\n    } else {\n      questionsParNiveau = [[2], [5], [15], [31], [2, 5, 15, 31]]\n    }\n    let QuestionsDisponibles = []\n    const IndiceNew = [0, 0, 0, 0, 0]\n    let NumQuestionsDisponibles\n    QuestionsDisponibles = gestionnaireFormulaireTexte({\n      saisie: this.sup,\n      min: 1,\n      max: 4,\n      defaut: 5,\n      melange: 5,\n      nbQuestions: this.nbQuestions,\n      shuffle: true\n    })\n    for (let i = 0; i < this.nbQuestions; i++) { // on a un tableau avec des strings : ['1', '1', '2']\n      NumQuestionsDisponibles = contraindreValeur(1, 5, parseInt(QuestionsDisponibles[i % QuestionsDisponibles.length]), 4) - 1\n      typesDeQuestionsDisponibles[i] = choice(questionsParNiveau[NumQuestionsDisponibles], typesDeQuestionsDisponibles.slice(IndiceNew[NumQuestionsDisponibles])) // Ce slice permet de gérer, par exemple, le mélange 1-1-2 pour 10 questions car il n'y a pas assez de choix différents pour le mélange 1.\n      if (typesDeQuestionsDisponibles[i] === undefined) { // Dans le cas, on a épuisé tous les choix différents d'un mélange\n        IndiceNew[NumQuestionsDisponibles] = i\n        typesDeQuestionsDisponibles[i] = choice(questionsParNiveau[NumQuestionsDisponibles], typesDeQuestionsDisponibles.slice(IndiceNew[NumQuestionsDisponibles]))\n      }\n    }\n    const listeTypeDeQuestions = combinaisonListesSansChangerOrdre(\n      typesDeQuestionsDisponibles,\n      this.nbQuestions\n    )\n\n    this.listeQuestions = [] // Liste de questions\n    this.listeCorrections = [] // Liste de questions corrigées\n    const droiteColor = (context.isHtml ? ['red', 'blue', 'green', 'black', 'magenta', '#f15929'] : ['black', 'black', 'black', 'black', 'black', 'black'])\n\n    for (let i = 0, texte, texteCorr, cpt = 0; i < this.nbQuestions && cpt < 50;) {\n      texte = ''\n      texteCorr = ''\n\n      const numDroites = shuffle([1, 2, 3, 4, 5])\n      // constRightNumbers = [1, 2, 3, 4, 5]\n      const d = []\n      const dE = []\n      const P = []\n      const objets = []\n      const objets2 = []\n      let code = []\n      let code2 = []\n\n      switch (listeTypeDeQuestions[i]) {\n      // \\n1: One step (0 to 3)\\n2: One step with distractor (4 to 6)\\n3: Two steps (9 to 15)\\n4: Three steps (19 to 31)\\n5 : Blend']\n        case 0: // si 1//2 et 2//3 alors 1//3\n          code = [[1, 2, 1], [2, 3, 1]]\n          break\n        case 1: // si 1//2 et 2T3 alors 1T3\n          code = [[1, 2, 1], [2, 3, -1]]\n          break\n        case 2: // si 1T2 et 2T3 alors 1//3\n          code = [[1, 2, -1], [2, 3, -1]]\n          break\n        case 3: // si 1T2 et 2//3 alors 1T3\n          code = [[1, 2, -1], [2, 3, 1]]\n          break\n        case 4: // si 1T2 et 2//3 alors 1T3 et 4 distracteur\n          code = [[1, 2, -1], [2, 3, 1]]\n          code2 = [[1, 4, 1]]\n          break\n        case 5: // si 1T2 et 2T3 alors 1//3 et 4 distracteur\n          code = [[1, 2, -1], [2, 3, -1]]\n          code2 = [[2, 4, 1]]\n          break\n        case 6: // si 1//2 et 2//3 alors 1//3 et 4 distracteur\n          code = [[1, 2, 1], [2, 3, 1]]\n          code2 = [[1, 4, -1]]\n          break\n        case 8: // Si 1//2 et 2//3 et 3//4 alors 1//4\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, 1]]\n          break\n        case 9: // Si 1//2 et 2//3 et 3T4 alors 1T4\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, -1]]\n          break\n        case 10: // Si 1//2 et 2T3 et 3//4 alors 1T4\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, 1]]\n          break\n        case 11: // Si 1//2 et 2T3 et 3T4 alors 1//4\n          code = [[1, 2, 1], [2, 3, -1], [3, 4, -1]]\n          break\n        case 12: // Si 1T2 et 2//3 et 3//4 alors 1T4\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, 1]]\n          break\n        case 13: // Si 1T2 et 2//3 et 3T4 alors 1//4\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, -1]]\n          break\n        case 14: // Si 1T2 et 2T3 et 3//4 alors 1//4\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, 1]]\n          break\n        case 15: // Si 1T2 et 2T3 et 3T4 alors 1T4\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, -1]]\n          break\n        case 16: // Si 1//2 et 2//3 et 3//4 et 4//5 alors 1//5\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, 1], [4, 5, 1]]\n          break\n        case 17: // Si 1//2 et 2//3 et 3T4 et 4//5 alors 1T5\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, -1], [4, 5, 1]]\n          break\n        case 18: // Si 1//2 et 2T3 et 3//4 et 4//5 alors 1T5\n          code = [[1, 2, 1], [2, 3, -1], [3, 4, 1], [4, 5, 1]]\n          break\n        case 19: // Si 1//2 et 2T3 et 3T4 et 4//5 alors 1//5\n          code = [[1, 2, 1], [2, 3, -1], [3, 4, -1], [4, 5, 1]]\n          break\n        case 20: // Si 1T2 et 2//3 et 3//4 et 4//5 alors 1T5\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, 1], [4, 5, 1]]\n          break\n        case 21: // Si 1T2 et 2//3 et 3T4 et 4//5 alors 1//5\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, -1], [4, 5, 1]]\n          break\n        case 22: // Si 1T2 et 2T3 et 3//4 et 4//5 alors 1//5\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, 1], [4, 5, 1]]\n          break\n        case 23: // Si 1T2 et 2T3 et 3T4 et 4//5 alors 1T5\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, -1], [4, 5, 1]]\n          break\n        case 24: // Si 1//2 et 2//3 et 3//4 et 4T5 alors 1T5\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, 1], [4, 5, -1]]\n          break\n        case 25: // Si 1//2 et 2//3 et 3T4 et 4T5 alors 1//5\n          code = [[1, 2, 1], [2, 3, 1], [3, 4, -1], [4, 5, -1]]\n          break\n        case 26: // Si 1//2 et 2T3 et 3//4 et 4T5 alors 1//5\n          code = [[1, 2, 1], [2, 3, -1], [3, 4, 1], [4, 5, -1]]\n          break\n        case 27: // Si 1//2 et 2T3 et 3T4 et 4T5 alors 1T5\n          code = [[1, 2, 1], [2, 3, -1], [3, 4, -1], [4, 5, -1]]\n          break\n        case 28: // Si 1T2 et 2//3 et 3//4 et 4T5 alors 1//5\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, 1], [4, 5, -1]]\n          break\n        case 29: // Si 1T2 et 2//3 et 3T4 et 4T5 alors 1T5\n          code = [[1, 2, -1], [2, 3, 1], [3, 4, -1], [4, 5, -1]]\n          break\n        case 30: // Si 1T2 et 2T3 et 3//4 et 4T5 alors 1T5\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, 1], [4, 5, -1]]\n          break\n        case 31: // Si 1T2 et 2T3 et 3T4 et 4T5 alors 1//5\n          code = [[1, 2, -1], [2, 3, -1], [3, 4, -1], [4, 5, -1]]\n          break\n      }\n\n      // mixed statement\n      const couleurd = []\n      const phrases = []\n      texte += 'We know that'\n      couleurd.push(randint(0, 5))\n      const codeAll = code.concat(code2)\n      for (let j = 0; j < codeAll.length; j++) {\n        let textetemp = `$(d_${numDroites[codeAll[j][0] - 1]})`\n        if (codeAll[j][2] === 1) {\n          textetemp += '||'\n          couleurd.push(couleurd[j])\n        } else {\n          textetemp += '\\\\perp'\n          couleurd.push((couleurd[j] + 1) % 6)\n        }\n        textetemp += `(d_${numDroites[codeAll[j][1] - 1]})$`\n        phrases.push(textetemp)\n      }\n      // sentences=shuffle(sentences)\n      for (let j = 0; j < codeAll.length - 1; j++) {\n        texte += phrases[j]\n        if (j !== codeAll.length - 2) texte += ','\n        else texte += ' And'\n      }\n      texte += phrases[codeAll.length - 1] + '.<br>'\n\n      // construction of the figure\n      context.fenetreMathalea2d = [-2, -2, 15, 10] // important avec la création des droites\n      const labels = []\n      P.push(point(0, 0))\n      let droiteP = droiteParPointEtPente(P[0], randint(-1, 1, -2) / 10, '', droiteColor[couleurd[0]])\n      droiteP.epaisseur = 2\n      droite.pointilles = false\n      d.push(droiteP)\n      const droiteE = droite(point(droiteP.x1, droiteP.y1), point(droiteP.x2, droiteP.y2), '')\n      droiteE.epaisseur = 2\n      dE.push(droiteE)\n      labels.push(labelOnLine(droiteE, `$(d_${numDroites[codeAll[0][0] - 1]})$`))\n      objets.push(d[0])\n      objets2.push(dE[0])\n      for (let x = 0; x < codeAll.length; x++) {\n        if (codeAll[x][2] === 1) {\n          P.push(point((x + 1) * 2, (x + 1) * 2))\n          droiteP = droiteParPointEtParallele(P[x + 1], d[codeAll[x][0] - 1], '', droiteColor[couleurd[x + 1]])\n          droiteP.epaisseur = 2\n          droiteP.pointilles = d[[codeAll[x][0] - 1]].pointilles\n          d.push(droiteP)\n          const droiteP2 = droite(point(droiteP.x1, droiteP.y1), point(droiteP.x2, droiteP.y2), '')\n          droiteP2.epaisseur = 2\n          dE.push(droiteP2)\n          labels.push(labelOnLine(droiteP2, `$(d_${numDroites[codeAll[x][1] - 1]})$`))\n        } else {\n          P.push(point((x + 1) * 2, (x + 1) * 2))\n          droiteP = droiteParPointEtPerpendiculaire(P[x + 1], d[codeAll[x][0] - 1], '', droiteColor[couleurd[x + 1]])\n          droiteP.epaisseur = 2\n          droiteP.pointilles = x % 3 + 1\n          d.push(droiteP)\n          const droiteP2 = droite(point(droiteP.x1, droiteP.y1), point(droiteP.x2, droiteP.y2), '')\n          labels.push(labelOnLine(droiteP2, `$(d_${numDroites[codeAll[x][1] - 1]})$`))\n          droiteP2.epaisseur = 2\n          dE.push(droiteP2)\n          const Inter = pointIntersectionDD(d[codeAll[x][0] - 1], droiteP)\n          const PP = rotation(P[x + 1], Inter, 90)\n          objets.push(codageAngleDroit(PP, Inter, P[x + 1], 'black', 0.6))\n          objets2.push(codageAngleDroit(PP, Inter, P[x + 1], 'black', 0.6))\n        }\n        objets.push(d[x + 1])\n        objets2.push(dE[x + 1])\n      }\n      objets2.push(...labels)\n      objets.push(...labels)\n\n      if (this.sup3) {\n        texte += (context.vue === 'diap' ? '<center>' : '') + mathalea2d({\n          xmin: -2,\n          xmax: 15,\n          ymin: -2,\n          ymax: 10,\n          pixelsParCm: 20,\n          scale: (context.vue !== 'latex' ? 0.3 : 0.6),\n          mainlevee: false,\n          amplitude: 0.3\n        }, objets2) + '<br>' + (context.vue === 'diap' ? '</center>' : '')\n      }\n      texte += `What can we say about $(d_${numDroites[code[0][0] - 1]})$ and $(d_${numDroites[code[code.length - 1][1] - 1]})$?`\n      if (context.isAmc && !this.sup3) {\n        texte += ' We can help ourselves by drawing a figure.'\n      }\n\n      // orderly reasoning correction\n      texteCorr = 'From the statement, we can create the following diagram (there are an infinite number of them).<br>'\n      if ([2, 5, 15, 31].indexOf(listeTypeDeQuestions[i]) === -1 && !this.sup2) {\n        texteCorr += ' The parallel lines given in the statement are the same'\n        texteCorr += (context.html) ? ' color/style.<br>' : 'style.<br>'\n      }\n      texteCorr += mathalea2d({\n        xmin: -2,\n        xmax: 15,\n        ymin: -2,\n        ymax: 10,\n        pixelsParCm: 20,\n        scale: (context.vue !== 'latex' ? 0.3 : 0.6),\n        mainlevee: false,\n        amplitude: 0.3\n      }, objets) + '<br>'\n      for (let j = 0; j < code.length - 1; j++) {\n        if (this.correctionDetaillee) texteCorr += 'We know that :'\n        else texteCorr += 'As'\n        texteCorr += `$(d_${numDroites[code[j][0] - 1]})`\n        if (code[j][2] === 1) texteCorr += '||'\n        else texteCorr += '\\\\perp'\n        texteCorr += `(d_${numDroites[code[j][1] - 1]})$ and`\n        texteCorr += `$(d_${numDroites[code[j + 1][0] - 1]})`\n        if (code[j + 1][2] === 1) texteCorr += '||'\n        else texteCorr += '\\\\perp'\n        texteCorr += `(d_${numDroites[code[j + 1][1] - 1]})$`\n        // what property?\n        if (code[j][2] * code[j + 1][2] === -1) { // Une parallèle et une perpendiculaire\n          if (this.correctionDetaillee) texteCorr += '.<br> Now “If two lines are parallel then any line perpendicular to one is also perpendicular to the other”.<br>So'\n          else texteCorr += ', we deduce that'\n          texteCorr += ` $(d_${numDroites[code[0][0] - 1]})\\\\perp(d_${numDroites[code[j + 1][1] - 1]})$.<br>`\n          code[j + 1][0] = code[j][0]\n          code[j + 1][2] = -1\n        } else if (code[j][2] > 0) { // deux parallèles\n          if (this.correctionDetaillee) texteCorr += '.<br> Or “If two lines are parallel to the same line then they are parallel to each other”.<br>So'\n          else texteCorr += ', we deduce that'\n          texteCorr += ` $(d_${numDroites[code[0][0] - 1]})||(d_${numDroites[code[j + 1][1] - 1]})$.<br>`\n          code[j + 1][0] = code[j][0]\n          code[j + 1][2] = 1\n        } else { // deux perpendiculaires\n          if (this.correctionDetaillee) texteCorr += '.<br> Or “If two lines are perpendicular to the same line then they are parallel to each other”.<br>So'\n          else texteCorr += ', we deduce that'\n          texteCorr += ` $(d_${numDroites[code[0][0] - 1]})||(d_${numDroites[code[j + 1][1] - 1]})$.<br>`\n          code[j + 1][0] = code[j][0]\n          code[j + 1][2] = 1\n        }\n      }\n\n      /** ********************** AMC Open *****************************/\n      this.autoCorrection[i] = {}\n      this.autoCorrection[i].options = { ordered: false }\n      this.autoCorrection[i].enonce = texte + '<br>'\n      this.autoCorrection[i].propositions = [\n        {\n          texte: texteCorr,\n          statut: 3\n        }\n      ]\n      /****************************************************/\n\n      if (this.listeQuestions.indexOf(texte) === -1) {\n      // If the question has never been asked, we create another one\n        this.listeQuestions.push(texte + '<br>')\n        this.listeCorrections.push(texteCorr + '<br>')\n        i++\n      }\n      cpt++\n    }\n\n    listeQuestionsToContenu(this)\n  }\n  this.besoinFormulaireTexte = ['Number of reasoning steps', 'Numbers separated by hyphens\\n1: One step\\n2: One step with distractor\\n3: Two steps\\n4: Three steps\\n5: Mix']\n  this.besoinFormulaire2CaseACocher = ['Only perpendiculars', false]\n  this.besoinFormulaire3CaseACocher = ['With the drawing', true]\n}\n\n/**\n * Ajouter une étiquette sur une droite.\n * @param {*} droite La droite où on va rajouter une étiquette\n * @param {*} nom Le nom de la droite doit être au format latex, c'est-à-dire compris entre $ et $\n * @param {*} options Les options permettant de personnaliser la position de l'étiquette et la mise en forme\n *  options.preferedPosition La position à privilégier si possible sur le bord de l\"image ('left\", 'right', 'above', 'below')\n *  options.usedPosition Un tableau des anciennes positions déjà allouées pour éviter les colisions avec des étiquettes d'autres droites\n *  options.taille La taille de la police de l'étiquette par défaut 6\n *  options.color La couleur de l\"default label 'red\"\n * @returns LatexParCoordonneesBox L'étiquette\n *\n * Exemple :\n *   context.fenetreMathalea2d = [xmin + 0.2, ymin, xmax, ymax] // important pour la position des labels\n *   const d3nom = labelOnLine(d3, '$' + noms[3] + '$', { color: 'blue', taille: 8, preferedPosition: 'left' })\n *   const d0nom = labelOnLine(d0, '$' + noms[0] + '$', { color: 'red', taille: 8, usedPosition: [d3nom] })\n *\n * @author Mickael Guironnet\n */\nexport function labelOnLine (droite, nom, options = {}) {\n  if (options.preferedPosition === undefined) options.preferedPosition = 'auto'\n  if (options.usedPosition === undefined) options.usedPosition = []\n  if (options.taille === undefined) options.taille = 6\n  if (options.color === undefined) options.color = 'red'\n\n  const debug = false\n\n  // const width = Math.ceil((name.replaceAll('$', '').length) * 10 * Math.log10(options.size))\n  const largeur = Math.ceil((nom.replaceAll('$', '').length) * options.taille * 10 / 12)\n  const hauteur = 20\n  let absNom, ordNom, leNom, anchor, usedPosition\n  const positions = []\n  if (nom !== '') {\n    if (egal(droite.b, 0, 0.05)) { // ax+c=0 x=-c/a est l'équation de la droite\n      // almost vertical line\n      absNom = -droite.c / droite.a + largeur * 0.5 / context.pixelsParCm + 2 / context.pixelsParCm\n      ordNom = context.fenetreMathalea2d[1] + 1 // l'ordonnée du label est ymin +1\n      anchor = 'right'\n      usedPosition = 'below'\n      leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n      positions.push({ label: leNom, position: usedPosition, anch: anchor })\n    } else if (egal(droite.a, 0, 0.05)) { // by+c=0 y=-c/b est l'équation de la droite\n      // almost horizontal straight line\n      absNom = context.fenetreMathalea2d[0] + 1 // l'abscisse du label est xmin +1\n      ordNom = -droite.c / droite.b + hauteur * 0.5 / context.pixelsParCm\n      anchor = 'above'\n      usedPosition = 'left'\n      leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n      positions.push({ label: leNom, position: usedPosition, anch: anchor })\n    } else { // a et b sont différents de 0 ax+by+c=0 est l'équation\n      // y=(-a.x-c)/b is the Cartesian equation and x=(-by-c)/a\n      const y0 = (-droite.a * (context.fenetreMathalea2d[0] + 1) - droite.c) / droite.b\n      const y1 = (-droite.a * (context.fenetreMathalea2d[2] - 1) - droite.c) / droite.b\n      const x0 = (-droite.b * (context.fenetreMathalea2d[1] + 1) - droite.c) / droite.a\n      const x1 = (-droite.b * (context.fenetreMathalea2d[3] - 1) - droite.c) / droite.a\n      if (y0 > context.fenetreMathalea2d[1] && y0 < context.fenetreMathalea2d[3]) {\n        // on the left: either below or below\n        absNom = context.fenetreMathalea2d[0] + 1\n        ordNom = y0 - droite.pente * (largeur * 0.5 / context.pixelsParCm) + (droite.pente > 0 ? -1 : 1) * hauteur * 0.5 / context.pixelsParCm\n        anchor = (droite.pente > 0 ? 'below' : 'above')\n        usedPosition = 'left'\n        if (ordNom < context.fenetreMathalea2d[1] + 1 || ordNom > context.fenetreMathalea2d[3] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[1] + 1) + '<' + ordNom + '<' + (context.fenetreMathalea2d[3] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n        // on the left: either below or below\n        absNom = context.fenetreMathalea2d[0] + 1\n        ordNom = y0 + droite.pente * (largeur * 0.5 / context.pixelsParCm) - (droite.pente > 0 ? -1 : 1) * hauteur * 0.5 / context.pixelsParCm\n        anchor = (droite.pente > 0 ? 'above' : 'below')\n        usedPosition = 'left'\n        if (ordNom < context.fenetreMathalea2d[1] + 1 || ordNom > context.fenetreMathalea2d[3] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[1] + 1) + '<' + ordNom + '<' + (context.fenetreMathalea2d[3] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n      }\n      if (y1 > context.fenetreMathalea2d[1] && y1 < context.fenetreMathalea2d[3]) {\n        // to the right\n        absNom = context.fenetreMathalea2d[2] - 1\n        ordNom = y1 - droite.pente * (largeur * 0.5 / context.pixelsParCm) + (droite.pente > 0 ? -1 : 1) * hauteur * 0.5 / context.pixelsParCm\n        anchor = (droite.pente > 0 ? 'below' : 'above')\n        usedPosition = 'right'\n        if (ordNom < context.fenetreMathalea2d[1] + 1 || ordNom > context.fenetreMathalea2d[3] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[1] + 1) + '<' + ordNom + '<' + (context.fenetreMathalea2d[3] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n      }\n      if (x0 > context.fenetreMathalea2d[0] && x0 < context.fenetreMathalea2d[2]) {\n        // bottom: either left or right\n        absNom = x0 + (droite.pente > 0 ? -1 : 1) * largeur * 0.5 / context.pixelsParCm - (droite.pente > 0 ? 1 : 1) * (hauteur * 0.5 / context.pixelsParCm) / droite.pente - (droite.pente > 0 ? 1 : -1) * 2 / context.pixelsParCm\n        ordNom = context.fenetreMathalea2d[1] + 1\n        anchor = (droite.pente > 0 ? 'left' : 'right')\n        usedPosition = 'below'\n        if (absNom < context.fenetreMathalea2d[0] + 1 || absNom > context.fenetreMathalea2d[2] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[0] + 1) + '<' + absNom + '<' + (context.fenetreMathalea2d[2] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n        // down on the other side\n        absNom = x0 - (droite.pente > 0 ? -1 : 1) * largeur * 0.5 / context.pixelsParCm + (droite.pente > 0 ? 1 : 1) * (hauteur * 0.5 / context.pixelsParCm) / droite.pente + (droite.pente > 0 ? 1 : -1) * 2 / context.pixelsParCm\n        ordNom = context.fenetreMathalea2d[1] + 1\n        anchor = (droite.pente > 0 ? 'right' : 'left')\n        usedPosition = 'below'\n        if (absNom < context.fenetreMathalea2d[0] + 1 || absNom > context.fenetreMathalea2d[2] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[0] + 1) + '<' + absNom + '<' + (context.fenetreMathalea2d[2] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n      }\n      if (x1 > context.fenetreMathalea2d[0] && x1 < context.fenetreMathalea2d[2]) {\n        // at the top: either left or right\n        absNom = x1 + (droite.pente > 0 ? -1 : 1) * largeur * 0.5 / context.pixelsParCm - (droite.pente > 0 ? 1 : 1) * (hauteur * 0.5 / context.pixelsParCm) / droite.pente - (droite.pente > 0 ? 1 : -1) * 2 / context.pixelsParCm\n        ordNom = context.fenetreMathalea2d[3] - 1\n        anchor = (droite.pente > 0 ? 'left' : 'right')\n        usedPosition = 'above'\n        if (absNom < context.fenetreMathalea2d[0] + 1 || absNom > context.fenetreMathalea2d[2] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[0] + 1) + '<' + absNom + '<' + (context.fenetreMathalea2d[2] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n        // at the top of the other side\n        absNom = x1 - (droite.pente > 0 ? -1 : 1) * largeur * 0.5 / context.pixelsParCm + (droite.pente > 0 ? 1 : 1) * (hauteur * 0.5 / context.pixelsParCm) / droite.pente + (droite.pente > 0 ? 1 : -1) * 2 / context.pixelsParCm\n        ordNom = context.fenetreMathalea2d[3] - 1\n        anchor = (droite.pente > 0 ? 'right' : 'left')\n        usedPosition = 'above'\n        if (absNom < context.fenetreMathalea2d[0] + 1 || absNom > context.fenetreMathalea2d[2] - 1) {\n          if (debug) console.log('problem:name:' + nom + ':position:' + usedPosition + (context.fenetreMathalea2d[0] + 1) + '<' + absNom + '<' + (context.fenetreMathalea2d[2] - 1))\n        } else {\n          leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n          positions.push({ label: leNom, position: usedPosition, anch: anchor })\n        }\n      }\n      let xgauche, xdroite\n      if (y0 > context.fenetreMathalea2d[1] && y0 < context.fenetreMathalea2d[3]) {\n        xgauche = context.fenetreMathalea2d[0]\n      } else {\n        xgauche = Math.min(x0, x1)\n      }\n      if (y1 > context.fenetreMathalea2d[1] && y1 < context.fenetreMathalea2d[3]) {\n        xdroite = context.fenetreMathalea2d[2]\n      } else {\n        xdroite = Math.max(x0, x1)\n      }\n      // between\n      absNom = (xgauche + xdroite) / 2\n      ordNom = pointSurDroite(droite, absNom).y\n      anchor = (droite.pente > 0 ? 'left' : 'right')\n      usedPosition = 'middle'\n      leNom = latexParCoordonneesBox(nom.substr(1, nom.length - 2), absNom, ordNom, options.color, largeur, hauteur, 'white', options.taille, { anchor })\n      positions.push({ label: leNom, position: usedPosition, anch: anchor })\n    }\n\n    // check if there are collisions between labels\n    for (let i = 0; i < positions.length; i++) {\n      if (positions[i].position === 'middle') continue\n      const coli = []\n      for (let j = 0; j < options.usedPosition.length; j++) {\n        const label = options.usedPosition[j]\n        const dis = segment(point(label.x, label.y), point(positions[i].label.x, positions[i].label.y)).longueur * context.pixelsParCm\n        // collision of two rectangles\n        const XYlabel = [label.x * context.pixelsParCm - label.largeur / 2, label.x * context.pixelsParCm + label.largeur / 2, label.y * context.pixelsParCm - label.hauteur / 2, label.y * context.pixelsParCm + label.hauteur / 2]\n        const XYlabel2 = [positions[i].label.x * context.pixelsParCm - positions[i].label.largeur / 2, positions[i].label.x * context.pixelsParCm + positions[i].label.largeur / 2, positions[i].label.y * context.pixelsParCm - positions[i].label.hauteur / 2, positions[i].label.y * context.pixelsParCm + positions[i].label.hauteur / 2]\n        if (debug) console.log('coli:name:' + nom + ':position:' + positions[i].position + ':i:' + i + ':j:' + j + ':say:' + dis.toFixed(2) + ':text:' + label.texte + ':XYlabel:' + XYlabel[0].toFixed(1) + ',' + XYlabel[1].toFixed(1) + ',' + XYlabel[2].toFixed(1) + ',' + XYlabel[3].toFixed(1) + ':XYlabel2:' + XYlabel2[0].toFixed(1) + ',' + XYlabel2[1].toFixed(1) + ',' + XYlabel2[2].toFixed(1) + ',' + XYlabel2[3].toFixed(1))\n        const colision = (XYlabel[0] < XYlabel2[1]) && (XYlabel[1] > XYlabel2[0]) && (XYlabel[2] < XYlabel2[3]) && (XYlabel[3] > XYlabel2[2])\n        // collision of two circles\n        const r0 = Math.max(label.largeur / 2, label.hauteur / 2)\n        const r1 = Math.max(positions[i].label.largeur / 2, positions[i].label.hauteur / 2)\n        let colision2 = true\n        if (dis > r0 + r1 || dis < Math.abs(r0 - r1)) colision2 = false\n        coli[j] = [dis, colision]\n        if (debug) console.log('coli:name:' + nom + ':position:' + positions[i].position + ':anchor:' + positions[i].anch + ':i:' + i + ':j:' + j + ':say:' + dis.toFixed(2) + 'text:' + label.texte + ':collision:' + (colision ? '1' : '0') + ':coli_cer:' + (colision2 ? '1' : '0'))\n      }\n      positions[i].colision = coli\n    }\n    // 1st strategy: user preference\n    // we only check if there is a collision\n    const found = [false, 0]\n    for (let i = 0; i < positions.length && !found[0]; i++) {\n      if (positions[i].position === 'middle') continue\n      if (positions[i].position === options.preferedPosition) {\n        found[0] = true\n        for (let j = 0; j < options.usedPosition.length; j++) {\n          if (positions[i].colision[j][1]) found[0] = false\n          if (debug) console.log('1st:name:' + nom + ':position:' + positions[i].position + ':i:' + i + ':j:' + j + ':collision:' + positions[i].colision[j][1] + ':preferredPosition:' + options.preferedPosition)\n        }\n        found[1] = i\n      }\n    }\n\n    // 2nd strategy: the furthest in terms of distance!\n    let disMax = [0, 0, 0]\n    for (let i = 0; i < positions.length && !found[0] && options.usedPosition.length > 0; i++) {\n      if (positions[i].position === 'middle') continue\n      let dis = [1000, 0, 0]\n      for (let j = 0; j < options.usedPosition.length; j++) {\n        if (positions[i].colision[j][0] < dis[0]) {\n          dis = [positions[i].colision[j][0], i, j]\n          if (debug) console.log('2nd:name:' + nom + ':position:' + positions[i].position + ':anchor:' + positions[i].anch + ':i:' + i + ':j:' + j + 'say:' + dis[0].toFixed(2) + ':collision:' + positions[i].colision[j][1])\n        }\n      }\n      if (dis[0] > disMax[0]) {\n        disMax = dis\n        if (debug) console.log('Max 2nd:name:' + nom + ':position:' + positions[i].position + ':anchor:' + positions[i].anch + ':i:' + i + 'say:' + dis[0].toFixed(2))\n      }\n    }\n    let colision = false\n    if (disMax[0] > 0) {\n      colision = positions[disMax[1]].colision[disMax[2]][1]\n      if (debug) console.log('Max end: 2nd:name:' + nom + ':position:' + positions[disMax[1]].position + ':anchor:' + positions[disMax[1]].anch + ':i:' + disMax[1] + ':j:' + disMax[2] + 'disMax:' + disMax[0] + ':collision:' + positions[disMax[1]].colision[disMax[2]][1])\n    }\n    // 1st: if preference then Ok otherwise furthest distance without overlap otherwise the first solution if no comparison otherwise the last middle\n    leNom = found[0] ? positions[found[1]].label : (disMax[0] > 0 && !colision ? positions[disMax[1]].label : positions[options.usedPosition.length === 0 ? 0 : positions.length - 1].label)\n  } else {\n    leNom = vide2d()\n  }\n  return 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