VN-TST 2023 Problem 3

Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$ . $BE, CF$ are the heights of $\triangle ABC$ , and $BE, CF$ intersect at $H$ . Let $M$ be the midpoint of $AH$ , and $K$ be the point on $EF$ such that $HK \perp EF$ . A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$ , $Q$ , respectively.

Show that the tangent line of $(CQE)$ at $E$ , the tangent line of $(BPF)$ at $F$ , and $MK$ concur.