## Triangles with non rational sides and integral area

### Problem 390

Published on Saturday, 23rd June 2012, 08:00 pm; Solved by 391; Difficulty rating: 60%Consider the triangle with sides √5, √65 and √68. It can be shown that this triangle has area 9.

S(n) is the sum of the areas of all triangles with sides √(1+b^{2}), √(1+c^{2}) and √(b^{2}+c^{2}) (for positive integers b and c ) that have an integral area not exceeding n.

The example triangle has b=2 and c=8.

S(10^{6})=18018206.

Find S(10^{10}).