ÖMO 1997 Problem 3

Let be given a triangle $ABC$ . Points $P$ on side $AC$ and $Y$ on the production of $CB$ beyond $B$ are chosen so that $Y$ subtends equal angles with $AP$ and $PC$ . Similarly, $Q$ on side $BC$ and $X$ on the production of $AC$ beyond $C$ are such that $X$ subtends equal angles with $BQ$ and $QC$ . Lines $YP$ and $XB$ meet at $R$ , $XQ$ and $YA$ meet at $S$ , and $XB$ and $YA$ meet at $D$ . Prove that $PQRS$ is a parallelogram if and only if $ACBD$ is a cyclic quadrilateral.