Problem 3

Two circles in a plane intersect. Let

A

be one of their points of intersection. Starting simultaneously from

A

, two points move with constant speeds, each travelling along its own circle in the same sense. The two points return to

A

simultaneously after one revolution. Prove that there exists a fixed point

P

in the plane such that, at any time, the distances from

P

to the moving points are equal.

Step 1 of 5: Name the centers and moving points

O,O\prime\text{ are the centers; }X,X\prime\text{ are the moving points}

Detailed analysis

Let O and O′ be the centers of the two circles, and let X and X′ be the moving points on them.