KMO 2021 Problem 1

Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$ . Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$ , respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$ , respectively. The perpendicular line to $AD$ , passing through $E, F$ meets $\Omega$ again at $G, H$ , respectively. Suppose that $AE$ and $GD$ meet at $P$ , $EH$ and $GF$ meet at $Q$ , and $HD$ and $AF$ meet at $R$ . Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$ .