TSTST 2012 Problem 2

Let $ABCD$ be a quadrilateral with $AC = BD$ . Diagonals $AC$ and $BD$ meet at $P$ . Let $\omega_1$ and $O_1$ denote the circumcircle and the circumcenter of triangle $ABP$ . Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$ . Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$ ), respectively. Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$ ) and $\widehat {TP}$ (not including $C$ ). Prove that $MN \parallel O_1O_2$ .