BY-MO 2001 Problem 3

Three distinct points $A$ , $B$ , and $N$ are marked on the line $l$ , with $B$ lying between $A$ and $N$ . For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$ , points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$ , $C$ , and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$ ; and $A$ , $B$ , $C$ , and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$ .