The following is a problem from curiosity, since I was looking at other inscription cases.
If I have two equally sized circles with centers P and Q with P != Q with radius r, and their intersection is not empty nor a single point, what is the largest possible square that can be constructed within that intersection?
I started off by drawing the line segment PQ as well as the perpendicular bisector passing intersection points A,B. Intuitively, I can draw a square with one of the diagonals being min(PQ,AB) but this doesn't seem to be the largest. My intuition after that is that I can make a square bigger if I choose opposite points on the arc between PA QB and PB QA respectively, but I can't prove it.