2017 OIM Problems/Problem 2

Problem

Let $ABC$ be a right triangle and $\Gamma$ its circumcircle. Let $D$ be a point on the segment $BC$ , distinct from $B$ and 4C $, and let$ M $be the midpoint of$ AD $. The line perpendicular to$ AB $passing through$ D $cuts$ AB $at$ E $and$ \Gamma $at$ F $, with point$ D $between$ E $and$ F $. The lines$ FC $and$ EM $intersect at the point$ X $. If$ \angle DAE = \angle AFE $, show that the line$ AX $is tangent to$ \Gamma$.