Side of a square

Question

How to find the side length of this type of squares if $A$ and $B$ are given

Answer 1

Use Power of a Point on the pair of intersecting chords in the smaller circle. If we call the half-side length $x$ (for simplicity sake), we have that $$(x - b)^{2} = x(x - a)$$ $$x^{2} - 2bx + b^{2} = x^{2} - ax$$ $$2bx - ax = b^{2}$$ $$x = \frac{b^{2}}{2b - a}.$$

The side length then is $$2x = \boxed{\frac{2b^{2}}{2b - a}}.$$