OME 1984 Problem 5

Let $A$ and $A'$ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$ . Find the locus of the midpoint of segment $BB'$ when $x$ varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.