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

## Description
<section id='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.
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>euler385()</code> should return 3776957309612154000.
    testString: assert.strictEqual(euler385(), 3776957309612154000);

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function euler385() {
  // Good luck!
  return true;
}

euler385();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```

</section>