BG-MO 1963 Problem 3

In the trapezium $ABCD$ , a point $M$ is chosen on the non-base segment $AB$ . Through the points $M,A,D$ and $M,B,C$ are drawn circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ . Prove that:

(a) the second intersection point $N$ of $k_1$ and $k_2$ lies on the other non-base segment $CD$ or on its continuation; (b) the length of the line $O_1O_2$ doesn't depend on the location of $M$ on $AB$ ; (c) the triangles $O_1MO_2$ and $DMC$ are similar. Find such a position of $M$ on $AB$ that makes $k_1$ and $k_2$ have the same radius.