Question 19
Let $O$ be the circumcircle of triangle $ABC$. The bisector of $\angle A$ meets $O$ again at $P$ and meets $BC$ at $D$. The perpendicular from $D$ to $AB$ meets $AB$ at $E$ and meets $O$ at $Q$, chosen so that $E$ lies on segment $DQ$. Let $F=PQ\cap BC$ and $G=AF\cap CQ$. Prove that $EG\parallel FC$.
