# Problem 385: Ellipses inside triangles

![graphic](img385.gif)

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.