---
id: 5900f3fc1000cf542c50ff0e
title: 'Problem 143: Investigating the Torricelli point of a triangle'
challengeType: 1
forumTopicId: 301772
dashedName: problem-143-investigating-the-torricelli-point-of-a-triangle
---

# --description--

Let ABC be a triangle with all interior angles being less than 120 degrees. Let X be any point inside the triangle and let $XA = p$, $XC = q$, and $XB = r$.

Fermat challenged Torricelli to find the position of X such that p + q + r was minimised.

Torricelli was able to prove that if equilateral triangles AOB, BNC and AMC are constructed on each side of triangle ABC, the circumscribed circles of AOB, BNC, and AMC will intersect at a single point, T, inside the triangle. Moreover he proved that T, called the Torricelli/Fermat point, minimises $p + q + r$. Even more remarkable, it can be shown that when the sum is minimised, $AN = BM = CO = p + q + r$ and that AN, BM and CO also intersect at T.

<img class="img-responsive center-block" alt="equilateral triangles AOB, BNC and AMC constructed on each side of triangle ABC; with the circumscribed circles of AOB, BNC, and AMC will intersecting at a single point, T, inside the triangle" src="https://cdn.freecodecamp.org/curriculum/project-euler/investigating-the-torricelli-point-of-a-triangle.png" style="background-color: white; padding: 10px;" />

If the sum is minimised and a, b, c, p, q and r are all positive integers we shall call triangle ABC a Torricelli triangle. For example, $a = 399$, $b = 455$, $c = 511$ is an example of a Torricelli triangle, with $p + q + r = 784$. Find the sum of all distinct values of $p + q + r ≤ 120000$ for Torricelli triangles.

# --hints--

`sumTorricelliTriangles()` should return `30758397`.

```js
assert.strictEqual(sumTorricelliTriangles(), 30758397);
```

# --seed--

## --seed-contents--

```js
function sumTorricelliTriangles() {

  return true;
}

sumTorricelliTriangles();
```

# --solutions--

```js
// solution required
```
