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 $C$ , 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 $\Gamma$ .