CAPS 2009 Problem 4

Given a circle, let $AB$ be a chord that is not a diameter, and let $C$ be a point on the longer arc $AB$ . Let $K$ and $L$ denote the reflections of $A$ and $B$ , respectively, about lines $BC$ and $AC$ , respectively. Prove that the distance between the midpoint of $AB$ and the midpoint of $KL$ is independent of the choice of $C$ .