Find the number of positive integer solutions such that $a+b+c\le 12$ and the shown square can be formed.
$a \perp b$ and $b\perp c$.
the segments $a,b,c$ lie completely inside the square as shown.
Here is my attempt but I am pretty sure this is not the efficient method
Let the angle between left edge of square and segment $a$ be $\alpha$. To form a square we need the horizontal projections equal the vertical projections. Using similar triangles it is easy to get to below equation $$\langle \cos\alpha ,~\sin \alpha\rangle \cdot \langle b-a-c,~a \rangle = 0 $$
I feel stuck after this. Any help ?