BY-MO 2007 Problem 2

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ , respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$ . Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$ . Both points move in clockwise direction and have the same linear velocity $v$ .

(a) Prove that all triangles $AM_1M_2$ are equilateral. (b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.