NL-BXMO-TST 2017 Problem 3

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$ . The midpoints of $AD$ and $AC$ are called $E$ and $F$ , respectively. Let $M$ be the circumcentre of $BEF$ . Prove that $AC$ and $BM$ are parallel.