Problem 3
Two circles in a plane intersect. Let be one of their points of intersection. Starting simultaneously from , two points move with constant speeds, each travelling along its own circle in the same sense. The two points return to simultaneously after one revolution. Prove that there exists a fixed point in the plane such that, at any time, the distances from to the moving points are equal.
Step 2 of 5: Construct the fixed point
Detailed analysis
Reflect A across the perpendicular bisector of OO′ and call the image P. This point is fixed throughout the motion.