ILL 1974 Problem 27

Let $C_1$ and $C_2$ be circles in the same plane, $P_1$ and $P_2$ arbitrary points on $C_1$ and $C_2$ respectively, and $Q$ the midpoint of segment $P_1P_2.$ Find the locus of points $Q$ as $P_1$ and $P_2$ go through all possible positions. Alternative version. Let $C_1, C_2, C_3$ be three circles in the same plane. Find the locus of the centroid of triangle $P_1P_2P_3$ as $P_1, P_2,$ and $P_3$ go through all possible positions on $C_1, C_2$ , and $C_3$ respectively.