BG-MO 1977 Problem 4

Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$ . Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$ . Prove that the locus of $F$ is an arc of a circle.

J. Tabov