VN-TST 2022 Problem 3

Let $ABCD$ be a parallelogram, $AC$ intersects $BD$ at $I$ . Consider point $G$ inside $\triangle ABC$ that satisfy $\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}$ . Let $E,G$ be projections of $C$ on $AG$ and $D$ on $BG$ . The $E-$ median line of $\triangle BEF$ and $F-$ median line of $\triangle AEF$ intersects at $H$ . $a)$ Prove that $AF,BE$ and $IH$ concurrent. Call the concurrent point $L$ . $b)$ Let $K$ be the intersection of $CE$ and $DF$ . Let $J$ circumcenter of $(LAB)$ and $M,N$ are respectively be circumcenters of $(EIJ)$ and $(FIJ)$ . Prove that $EM,FN$ and the line go through circumcenters of $(GAB),(KCD)$ are concurrent.