AHSME 1951 Problem 46

$AB$ is a fixed diameter of a circle whose center is $O$. From $C$, any point on the circle, a chord $CD$ is drawn perpendicular to $AB$. Then, as $C$ moves over a semicircle, the bisector of angle $OCD$ cuts the circle in a point that always: $\textbf{(A)}\ \text{bisects the arc }AB \textbf{(B)}\ \text{trisects the arc }AB \textbf{(C)}\ \text{varies}$ $\textbf{(D)}\ \text{is as far from }AB\text{ as from }D \textbf{(E)}\ \text{is equidistant from }B\text{ and }C$