Geometry Problem 990: Triangle, Incircle, Median, Cevian, Central Angle, Congruence, Circle, Secant, Midpoint, Parallel Lines.

In a triangle ABC the median BM intersects the incircle of center I at D and E, respectively (see the figure). DJ is parallel to AC and angles DIH and JIE are equal (H and J on the incircle). BH and BJ extended meet AC at F and G, respectively. Prove that AF = GC.