Question 172
Let $ABC$ be a triangle and let $\omega$ be its circumcircle. Let $\ell_{B}$ and $\ell_{C}$ be two parallel lines passing through $B$ and $C$ respectively. The lines $\ell_{B}$ and $\ell_{C}$ intersect with $\omega$ for the second time at the points $D$ and $E$ respectively, with $D$ belonging on the arc $AB$, and $E$ on the arc $AC$. Suppose that $DA$ intersects $\ell_{C}$ at $F$, and $EA$ intersects $\ell_{B}$ at $G$. If $O, O_{1}$ and $O_{2}$ are the circumcenters of the triangles $ABC, ADG$ and $AEF$ respectively, and $P$ is the center of the circumcircle of the triangle $OO_{1}O_{2}$, prove that $OP$ is parallel to $\ell_{B}$ and $\ell_{C}$.