Question 19
Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be midpoints of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ are circumcircles of triangles $ADP$, $BCP$ and $OMN$, respectively. Let $E$ and $F$ be the intersection point of $\omega_1$ and $\omega_3$ which is not on the arc $APD$ of $\omega_1$, and the intersection point of $\omega_2$ and $\omega_3$ which is not on the arc $BPC$ of $\omega_2$, respectively. Prove that $OE = OF$.