---
id: 5900f4971000cf542c50ffaa
title: 'Problem 299: Three similar triangles'
challengeType: 5
forumTopicId: 301951
dashedName: problem-299-three-similar-triangles
---

# --description--

Four points with integer coordinates are selected:

$A(a, 0)$, $B(b, 0)$, $C(0, c)$ and $D(0, d)$, with $0 < a < b$ and $0 < c < d$.

Point $P$, also with integer coordinates, is chosen on the line $AC$ so that the three triangles $ABP$, $CDP$ and $BDP$ are all similar.

<img class="img-responsive center-block" alt="points A, B, C, D and P creating three triangles: ABP, CDP, and BDP" src="https://cdn.freecodecamp.org/curriculum/project-euler/three-similar-triangles.gif" style="background-color: white; padding: 10px;">

It is easy to prove that the three triangles can be similar, only if $a = c$.

So, given that $a = c$, we are looking for triplets ($a$, $b$, $d$) such that at least one point $P$ (with integer coordinates) exists on $AC$, making the three triangles $ABP$, $CDP$ and $BDP$ all similar.

For example, if $(a, b, d) = (2, 3, 4)$, it can be easily verified that point $P(1, 1)$ satisfies the above condition. Note that the triplets (2,3,4) and (2,4,3) are considered as distinct, although point $P(1, 1)$ is common for both.

If $b + d &lt; 100$, there are 92 distinct triplets ($a$, $b$, $d$) such that point $P$ exists.

If $b + d &lt; 100\\,000$, there are 320471 distinct triplets ($a$, $b$, $d$) such that point $P$ exists.

If $b + d &lt; 100\\,000\\,000$, how many distinct triplets ($a$, $b$, $d$) are there such that point $P$ exists?

# --hints--

`threeSimilarTriangles()` should return `549936643`.

```js
assert.strictEqual(threeSimilarTriangles(), 549936643);
```

# --seed--

## --seed-contents--

```js
function threeSimilarTriangles() {

  return true;
}

threeSimilarTriangles();
```

# --solutions--

```js
// solution required
```