Distinct integer solutions to $a^2+b^2=c^2+d^2$ I'm trying to find integer solutions to
$$a^2+b^2=c^2+d^2$$
with values  $a> c > d > b>0$
Or in other words, two triangles with integer legs and equal hypotenuse lengths, not necessarily integer. Seems like a Diophantine equation to me, but I only learned how to solve Diophantine equations in the form of Pell's equations. I couldn't find anything on this equation when I checked Wikipedia. It is similar to a Pythagorean quadruple, although not quite, so that's not helpful either. How do I find integer solutions to this?

One way to find some solutions is to rewrite as $a^2-c^2=d^2-b^2$, which in turn becomes $(a-c)(a+c)=(d-b)(d+b)$.
Then choose a number that factors in more than one way and see what you get.
Example: $15=3\cdot 5=1\cdot 15$
Then $15=(4-1)(4+1)=(8-7)(8+7)$.
So take $a=4, b=7, c=1, d=8$
