Let a denote the length of the square side. Then the vertices have the coordinates:
A(0, 0), B(a, 0), C(a, a), D(0, a)
2. The circles around A, B and C intersect at the trough. You'll get the system of non-linear equations:
3. You'll get two solutions. The most plausible one is:
(x, y, a)=( 36.22862195, 34.45993254, 56.53903920)
4. My results look very strange so please check my calculations.