## Ellipses inside triangles

### Problem 385

Published on Sunday, 20th May 2012, 05:00 am; Solved by 189; Difficulty rating: 70%
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 **foci**^{1} 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).

^{1}The **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.