---
id: 5900f3d21000cf542c50fee5
title: 'Problem 102: Triangle containment'
challengeType: 1
forumTopicId: 301726
dashedName: problem-102-triangle-containment
---

# --description--

Three distinct points are plotted at random on a Cartesian plane, for which -1000 ≤ x, y ≤ 1000, such that a triangle is formed.

Consider the following two triangles:

```js
const exampleTriangles = [
  [[-340, 495], [-153, -910], [835, -947]],
  [[-175, 41], [-421, -714], [574, -645]]
];
```

It can be verified that first triangle contains the origin, whereas second triangle does not.

Using the `triangles` array containing coordinates of triangles, find the number of triangles for which the interior contains the origin.

# --hints--

`triangleContainment(exampleTriangles)` should return a number.

```js
assert(typeof triangleContainment(_exampleTriangles) === 'number');
```

`triangleContainment(exampleTriangles)` should return `1`.

```js
assert.strictEqual(triangleContainment(_exampleTriangles), 1);
```

`triangleContainment(testTriangles1)` should return `19`.

```js
assert.strictEqual(triangleContainment(_testTriangles1), 19);
```

`triangleContainment(testTriangles2)` should return `228`.

```js
assert.strictEqual(triangleContainment(_testTriangles2), 228);
```

# --seed--

## --after-user-code--

```js
const _exampleTriangles = [
  [[-340, 495], [-153, -910], [835, -947]],
  [[-175, 41], [-421, -714], [574, -645]]
];
const _testTriangles1 = [
  [[-866,121],[-928,358],[459,-843]],[[-568,-631],[-352,-580],[-349,189]],[[-737,849],[-963,-486],[-662,970]],[[135,334],[-967,-71],[-365,-792]],[[789,21],[-227,51],[990,-275]],[[240,412],[-886,230],[591,256]],[[-609,472],[-853,-754],[959,661]],[[401,521],[521,314],[929,982]],[[-499,784],[-208,71],[-302,296]],[[-557,-948],[-553,-526],[-864,793]],[[270,-626],[828,44],[37,14]],[[-412,224],[617,-593],[502,699]],[[41,-908],[81,562],[-849,163]],[[165,917],[761,-197],[331,-341]],[[-687,314],[799,755],[-969,648]],[[-164,25],[578,439],[-334,-576]],[[213,535],[874,-177],[-551,24]],[[-689,291],[-795,-225],[-496,-125]],[[465,461],[558,-118],[-568,-909]],[[567,660],[-810,46],[-485,878]],[[-147,606],[685,-690],[-774,984]],[[568,-886],[-43,854],[-738,616]],[[-800,386],[-614,585],[764,-226]],[[-518,23],[-225,-732],[-79,440]],[[-173,-291],[-689,636],[642,-447]],[[-598,-16],[227,410],[496,211]],[[-474,-930],[-656,-321],[-420,36]],[[-435,165],[-819,555],[540,144]],[[-969,149],[828,568],[394,648]],[[65,-848],[257,720],[-625,-851]],[[981,899],[275,635],[465,-877]],[[80,290],[792,760],[-191,-321]],[[-605,-858],[594,33],[706,593]],[[585,-472],[318,-35],[354,-927]],[[-365,664],[803,581],[-965,-814]],[[-427,-238],[-480,146],[-55,-606]],[[879,-193],[250,-890],[336,117]],[[-226,-322],[-286,-765],[-836,-218]],[[-913,564],[-667,-698],[937,283]],[[872,-901],[810,-623],[-52,-709]],[[473,171],[717,38],[-429,-644]],[[225,824],[-219,-475],[-180,234]],[[-530,-797],[-948,238],[851,-623]],[[85,975],[-363,529],[598,28]],[[-799,166],[-804,210],[-769,851]],[[-687,-158],[885,736],[-381,-461]],[[447,592],[928,-514],[-515,-661]],[[-399,-777],[-493,80],[-544,-78]],[[-884,631],[171,-825],[-333,551]],[[191,268],[-577,676],[137,-33]],[[212,-853],[709,798],[583,-56]],[[-908,-172],[-540,-84],[-135,-56]],[[303,311],[406,-360],[-240,811]],[[798,-708],[824,59],[234,-57]],[[491,693],[-74,585],[-85,877]],[[509,-65],[-936,329],[-51,722]],[[-122,858],[-52,467],[-77,-609]],[[850,760],[547,-495],[-953,-952]],[[-460,-541],[890,910],[286,724]],[[-914,843],[-579,-983],[-387,-460]],[[989,-171],[-877,-326],[-899,458]],[[846,175],[-915,540],[-1000,-982]],[[-852,-920],[-306,496],[530,-18]],[[338,-991],[160,85],[-455,-661]],[[-186,-311],[-460,-563],[-231,-414]],[[-932,-302],[959,597],[793,748]],[[-366,-402],[-788,-279],[514,53]],[[-940,-956],[447,-956],[211,-285]],[[564,806],[-911,-914],[934,754]],[[575,-858],[-277,15],[409,-714]]
];
const _testTriangles2 = [
  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[-292,-768],[-264,876]],[[-127,506],[-891,-331],[-744,-430]],[[778,584],[-750,-129],[-479,-94]],[[-876,-771],[-987,-757],[180,-641]],[[-777,-694],[411,-87],[329,190]],[[-347,-999],[-882,158],[-754,232]],[[-105,918],[188,237],[-110,-591]],[[-209,703],[-838,77],[838,909]],[[-995,-339],[-762,750],[860,472]],[[185,271],[-289,173],[811,-300]],[[2,65],[-656,-22],[36,-139]],[[765,-210],[883,974],[961,-905]],[[-212,295],[-615,-840],[77,474]],[[211,-910],[-440,703],[-11,859]],[[-559,-4],[-196,841],[-277,969]],[[-73,-159],[-887,126],[978,-371]],[[-569,633],[-423,-33],[512,-393]],[[503,143],[-383,-109],[-649,-998]],[[-663,339],[-317,-523],[-2,596]],[[690,-380],[570,378],[-652,132]],[[72,-744],[-930,399],[-525,935]],[[865,-983],[115,37],[995,826]],[[594,-621],[-872,443],[188,-241]],[[-1000,291],[754,234],[-435,-869]],[[-868,901],[654,-907],[59,181]],[[-868,-793],[-431,596],[-446,-564]],[[900,-944],[-680,-796],[902,-366]],[[331,430],[943,853],[-851,-942]],[[315,-538],[-354,-909],[139,721]],[[170,-884],[-225,-818],[-808,-657]],[[-279,-34],[-533,-871],[-972,552]],[[691,-986],[-800,-950],[654,-747]],[[603,988],[899,841],[-630,591]],[[876,-949],[809,562],[602,-536]],[[-693,363],[-189,495],[738,-1000]],[[-383,431],[-633,297],[665,959]],[[-740,686],[-207,-803],[188,-520]],[[-820,226],[31,-339],[10,121]],[[-312,-844],[624,-516],[483,621]],[[-822,-529],[69,-278],[800,328]],[[834,-82],[-759,420],[811,-264]],[[-960,-240],[-921,561],[173,46]],[[-324,909],[-790,-814],[-2,-785]],[[976,334],[-290,-891],[704,-581]],[[150,-798],[689,-823],[237,-639]],[[-551,-320],[876,-502],[-622,-628]],[[-136,845],[904,595],[-702,-261]],[[-857,-377],[-522,-101],[-943,-805]],[[-682,-787],[-888,-459],[-752,-985]],[[-571,-81],[623,-133],[447,643]],[[-375,-158],[72,-387],[-324,-696]],[[-660,-650],[340,188],[569,526]],[[727,-218],[16,-7],[-595,-988]],[[-966,-684],[802,-783],[-272,-194]],[[115,-566],[-888,47],[712,180]],[[-237,-69],[45,-272],[981,-812]],[[48,897],[439,417],[50,325]],[[348,616],[180,254],[104,-784]],[[-730,811],[-548,612],[-736,790]],[[138,-810],[123,930],[65,865]],[[-768,-299],[-49,-895],[-692,-418]],[[487,-531],[802,-159],[-12,634]],[[808,-179],[552,-73],[470,717]],[[720,-644],[886,-141],[625,144]],[[-485,-505],[-347,-244],[-916,66]],[[600,-565],[995,-5],[324,227]],[[-771,-35],[904,-482],[753,-303]],[[-701,65],[426,-763],[-504,-479]],[[409,733],[-823,475],[64,718]],[[865,975],[368,893],[-413,-433]],[[812,-597],[-970,819],[813,624]],[[193,-642],[-381,-560],[545,398]],[[711,28],[-316,771],[717,-865]],[[-509,462],[809,-136],[786,635]],[[618,-49],[484,169],[635,547]],[[-747,685],[-882,-496],[-332,82]],[[-501,-851],[870,563],[290,570]],[[-279,-829],[-509,397],[457,816]],[[-508,80],[850,-188],[483,-326]],[[860,-100],[360,119],[-205,787]],[[-870,21],[-39,-827],[-185,932]],[[826,284],[-136,-866],[-330,-97]],[[-944,-82],[745,899],[-97,365]],[[929,262],[564,632],[-115,632]],[[244,-276],[713,330],[-897,-214]],[[-890,-109],[664,876],[-974,-907]],[[716,249],[816,489],[723,141]],[[-96,-560],[-272,45],[-70,645]],[[762,-503],[414,-828],[-254,-646]],[[909,-13],[903,-422],[-344,-10]],[[658,-486],[743,545],[50,674]],[[-241,507],[-367,18],[-48,-241]],[[886,-268],[884,-762],[120,-486]],[[-412,-528],[879,-647],[223,-393]],[[851,810],[234,937],[-726,797]],[[-999,942],[839,-134],[-996,-189]],[[100,979],[-527,-521],[378,800]],[[544,-844],[-832,-530],[-77,-641]],[[43,889],[31,442],[-934,-503]],[[-330,-370],[-309,-439],[173,547]],[[169,945],[62,-753],[-542,-597]],[[208,751],[-372,-647],[-520,70]],[[765,-840],[907,-257],[379,918]],[[334,-135],[-689,730],[-427,618]],[[137,-508],[66,-695],[78,169]],[[-962,-123],[400,-417],[151,969]],[[328,689],[666,427],[-555,-642]],[[-907,343],[605,-341],[-647,582]],[[-667,-363],[-571,818],[-265,-399]],[[525,-938],[904,898],[725,692]],[[-176,-802],[-858,-9],[780,275]],[[580,170],[-740,287],[691,-97]],[[365,557],[-375,361],[-288,859]],[[193,737],[842,-808],[520,282]],[[-871,65],[-799,836],[179,-720]],[[958,-144],[744,-789],[797,-48]],[[122,582],[662,912],[68,757]],[[595,241],[-801,513],[388,186]],[[-103,-677],[-259,-731],[-281,-857]],[[921,319],[-696,683],[-88,-997]],[[775,200],[78,858],[648,768]],[[316,821],[-763,68],[-290,-741]],[[564,664],[691,504],[760,787]],[[694,-119],[973,-385],[309,-760]],[[777,-947],[-57,990],[74,19]],[[971,626],[-496,-781],[-602,-239]],[[-651,433],[11,-339],[939,294]],[[-965,-728],[560,569],[-708,-247]]
];
```

## --seed-contents--

```js
function triangleContainment(triangles) {

  return true;
}

const testTriangles1 = [
  [[-866,121],[-928,358],[459,-843]],[[-568,-631],[-352,-580],[-349,189]],[[-737,849],[-963,-486],[-662,970]],[[135,334],[-967,-71],[-365,-792]],[[789,21],[-227,51],[990,-275]],[[240,412],[-886,230],[591,256]],[[-609,472],[-853,-754],[959,661]],[[401,521],[521,314],[929,982]],[[-499,784],[-208,71],[-302,296]],[[-557,-948],[-553,-526],[-864,793]],[[270,-626],[828,44],[37,14]],[[-412,224],[617,-593],[502,699]],[[41,-908],[81,562],[-849,163]],[[165,917],[761,-197],[331,-341]],[[-687,314],[799,755],[-969,648]],[[-164,25],[578,439],[-334,-576]],[[213,535],[874,-177],[-551,24]],[[-689,291],[-795,-225],[-496,-125]],[[465,461],[558,-118],[-568,-909]],[[567,660],[-810,46],[-485,878]],[[-147,606],[685,-690],[-774,984]],[[568,-886],[-43,854],[-738,616]],[[-800,386],[-614,585],[764,-226]],[[-518,23],[-225,-732],[-79,440]],[[-173,-291],[-689,636],[642,-447]],[[-598,-16],[227,410],[496,211]],[[-474,-930],[-656,-321],[-420,36]],[[-435,165],[-819,555],[540,144]],[[-969,149],[828,568],[394,648]],[[65,-848],[257,720],[-625,-851]],[[981,899],[275,635],[465,-877]],[[80,290],[792,760],[-191,-321]],[[-605,-858],[594,33],[706,593]],[[585,-472],[318,-35],[354,-927]],[[-365,664],[803,581],[-965,-814]],[[-427,-238],[-480,146],[-55,-606]],[[879,-193],[250,-890],[336,117]],[[-226,-322],[-286,-765],[-836,-218]],[[-913,564],[-667,-698],[937,283]],[[872,-901],[810,-623],[-52,-709]],[[473,171],[717,38],[-429,-644]],[[225,824],[-219,-475],[-180,234]],[[-530,-797],[-948,238],[851,-623]],[[85,975],[-363,529],[598,28]],[[-799,166],[-804,210],[-769,851]],[[-687,-158],[885,736],[-381,-461]],[[447,592],[928,-514],[-515,-661]],[[-399,-777],[-493,80],[-544,-78]],[[-884,631],[171,-825],[-333,551]],[[191,268],[-577,676],[137,-33]],[[212,-853],[709,798],[583,-56]],[[-908,-172],[-540,-84],[-135,-56]],[[303,311],[406,-360],[-240,811]],[[798,-708],[824,59],[234,-57]],[[491,693],[-74,585],[-85,877]],[[509,-65],[-936,329],[-51,722]],[[-122,858],[-52,467],[-77,-609]],[[850,760],[547,-495],[-953,-952]],[[-460,-541],[890,910],[286,724]],[[-914,843],[-579,-983],[-387,-460]],[[989,-171],[-877,-326],[-899,458]],[[846,175],[-915,540],[-1000,-982]],[[-852,-920],[-306,496],[530,-18]],[[338,-991],[160,85],[-455,-661]],[[-186,-311],[-460,-563],[-231,-414]],[[-932,-302],[959,597],[793,748]],[[-366,-402],[-788,-279],[514,53]],[[-940,-956],[447,-956],[211,-285]],[[564,806],[-911,-914],[934,754]],[[575,-858],[-277,15],[409,-714]]
];

triangleContainment(testTriangles1);
```

# --solutions--

```js
function triangleContainment(triangles) {
  function isInTriangle(triangle, point) {
    // Based on https://stackoverflow.com/a/14382692
    const area = triangleArea(triangle);
    const s = getS(area, point, triangle);
    const t = getT(area, point, triangle);

    if (s > 0 && t > 0 && 1 - s - t > 0) {
      return true;
    }
    return false;
  }

  function getS(area, point, triangle) {
    return (
      (1 / (2 * area)) *
      (triangle.A.y * triangle.C.x -
        triangle.A.x * triangle.C.y +
        point.x * (triangle.C.y - triangle.A.y) +
        point.y * (triangle.A.x - triangle.C.x))
    );
  }

  function getT(area, point, triangle) {
    return (
      (1 / (2 * area)) *
      (triangle.A.x * triangle.B.y -
        triangle.A.y * triangle.B.x +
        point.x * (triangle.A.y - triangle.B.y) +
        point.y * (triangle.B.x - triangle.A.x))
    );
  }

  function triangleArea(triangle) {
    return (
      0.5 *
      (-triangle.B.y * triangle.C.x +
        triangle.A.y * (-triangle.B.x + triangle.C.x) +
        triangle.A.x * (triangle.B.y - triangle.C.y) +
        triangle.B.x * triangle.C.y)
    );
  }

  function parseTriangle(triangle) {
    const points = [];
    for (let i = 0; i < triangle.length; i++) {
      const [coordinateX, coordinateY] = triangle[i];
      points.push({ x: coordinateX, y: coordinateY });
    }
    const [pointA, pointB, pointC] = points;
    return {
      A: pointA,
      B: pointB,
      C: pointC
    };
  }

  const origin = { x: 0, y: 0 };

  let trianglesInside = 0;
  for (let i = 0; i < triangles.length; i++) {
    const triangle = parseTriangle(triangles[i]);

    if (isInTriangle(triangle, origin)) {
      trianglesInside++;
    }
  }
  return trianglesInside;
}
```
