Special Squares

Two segments $AB$ and $CD$ of length $a$ are joined by a perpendicular $BC$ of length $b$
The segment $AD$ is the diagonal of a square of side $c$

For instance, is $a=23$ and $b=94$, then $c$ is an integer $c=74$
In the picture, points B and C fall inside the square, but it is not necessarily the case in general.

How many coprime pairs (a,b) produce an integer side square where the side $c\le {10}^7$ ?

Answer format: $count,\sum a,\sum b$

You are given: 161,36737,73474 for a threshold of 1000

My timing: 1 sec