Question 123
Let $P$ and $Q$ be distinct intersection points of circles $\omega_1$ and $\omega_2$. A line passing through point $P$ intersects $\omega_1$ again at point $A$ and $\omega_2$ again at point $B$. Line $l$ intersects $\omega_1$ at points $E$ and $D$, and intersects $\omega_2$ at points $C$ and $F$, where $C$ and $D$ are between points $E$ and $F$. ($A, B, C, D, E$, and $F$ are all distinct points.) Let $H$ be the intersection of $AE$ and $BC$, and $G$ be the intersection of $AD$ and $BF$. The line passing through point $C$, parallel to $BF$, and line passing through point $D$, parallel to $AE$, intersect at point $M$. Prove that the points $H, M$, and $G$ are collinear.
(Khulan Tumenbayar)
(Khulan Tumenbayar)