VMO 2022 Problem 3

Let $ABC$ be a triangle. Point $E,F$ moves on the opposite ray of $BA,CA$ such that $BF=CE$ . Let $M,N$ be the midpoint of $BE,CF$ . $BF$ cuts $CE$ at $D$ a) Suppost that $I$ is the circumcenter of $(DBE)$ and $J$ is the circumcenter of $(DCF)$ , Prove that $MN \parallel IJ$ b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$ . Prove that when $E$ varies on the opposite ray of $BA$ , $HK$ go through a fixed point