Question 30
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 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 $MN \parallel O_1O_2$.
(This problem was suggested by Steve Dinh.)
(This problem was suggested by Steve Dinh.)