Consider a parallelogram $WXYZ$, with points $A$ and $B$ on sides $WX$ and $XY$ respectively, so that $\angle WAZ = \angle YBZ$. Let the midpoint of $WY$ be $M$. Prove that $OM$, where $O$ is the centre of the circle $AXB$, is perpendicular to $WY$.
EDIT: In response to Mick's solution. I think you need to explain why the equal angles means the two lines are parallel. I think your solution breaks down when you say that KMN is a straight line without proof. Here is a picture where your first two paragraphs are correct, but doesn't solve the problem because the original angles are not the same.