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Triangles with non rational sides and integral area

Published on Saturday, 23rd June 2012, 08:00 pm; Solved by 502;
Difficulty rating: 60%

Problem 390

Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$. It can be shown that this triangle has area $9$.

$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (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 $S(10^{10})$.