CN-TST 2018 Problem 17

Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$ , and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$ . Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$ , respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$ , so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$ . Prove that $DF,OM,EG$ are concurrent.