VN-TST 2021 Problem 5

Given a fixed circle $(O)$ and two fixed points $B, C$ on that circle, let $A$ be a moving point on $(O)$ such that $\triangle ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$ and let $AD, BE, CF$ be the three heights of $\triangle ABC$ . In two rays $\overrightarrow{FA}, \overrightarrow{EA}$ , we pick respectively $M,N$ such that $FM = CE, EN = BF$ . Let $L$ be the intersection of $MN$ and $EF$ , and let $G \neq L$ be the second intersection of $(LEN)$ and $(LFM)$ .

a) Show that the circle $(MNG)$ always goes through a fixed point.

b) Let $AD$ intersects $(O)$ at $K \neq A$ . In the tangent line through $D$ of $(DKI)$ , we pick $P,Q$ such that $GP \parallel AB, GQ \parallel AC$ . Let $T$ be the center of $(GPQ)$ . Show that $GT$ always goes through a fixed point.