KöMaL 2015 C1323

Let $T$ denote the intersection of side $BC$ with the angle bisector drawn from vertex $A$ of a right-angled triangle. Let $F$ denote the midpoint of side $BC$ , and let $M$ be the intersection of the perpendicular bisector drawn at $F$ with another side. Given that the quadrilateral $ATFM$ is a kite, determine the angles of the triangle. ($A$ may denote any vertex of the triangle.)