OMCC 2006 Problem 2

Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$ , respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$ , $C$ is the second point of intersection of $AB$ and $\Gamma'$ , and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$ .