I am working on Project Euler 390.
The question is about triangles, and finding the area of a triangle with sides $\sqrt{a^2+1}, \sqrt{b^2+1}$ and $\sqrt{a^2+b^2}$, with $a, b \in \mathbb{Z}$. I have narrowed the problem down to solving the equation
$x^2 \cdot y^2 + x^2 + y^2 = (2\cdot c)^2 \text{ with } x, y, c \in \mathbb{Z}^+$
This is not a problem for $c \le 10^6$ (brute force), but I have to calculate up to $10^{10}$. I would like to know how to solve these kind of equations, without any brute force attack. I have searched for a few days on Google, but the general solutions to the Diophantine equations I found were never appliable to my problem.
Any suggestions are welcome (even the name of this kind of equation), although I would appreciate not being told the answer to the problem.