---
id: 5900f4ee1000cf542c510000
title: 'Problem 385: Ellipses inside triangles'
challengeType: 5
forumTopicId: 302049
dashedName: problem-385-ellipses-inside-triangles
---

# --description--

For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside T.

For a given n, consider triangles T such that:

-   the vertices of T have integer coordinates with absolute value ≤ n, and
-   the foci1 of the largest-area ellipse inside T are (√13,0) and (-√13,0).

Let A(n) be the sum of the areas of all such triangles.

For example, if n = 8, there are two such triangles. Their vertices are (-4,-3),(-4,3),(8,0) and (4,3),(4,-3),(-8,0), and the area of each triangle is 36. Thus A(8) = 36 + 36 = 72.

It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.

Find A(1 000 000 000).

1The foci (plural of focus) of an ellipse are two points A and B such that for every point P on the boundary of the ellipse, AP + PB is constant.

# --hints--

`euler385()` should return 3776957309612154000.

```js
assert.strictEqual(euler385(), 3776957309612154000);
```

# --seed--

## --seed-contents--

```js
function euler385() {

  return true;
}

euler385();
```

# --solutions--

```js
// solution required
```