Geometry Problem 991: Triangle, Incircle, Tangency Points, Isosceles, Midpoint, Collinearity, Congruence, Circle. Level: School, College

The figure shows a triangle ABC of incenter I and BD = AB (D on AC). The incircle of triangle BCD is tangent to CD and BD at E and F, respectively. If M is the midpoint of DI, prove that points E, M, and F are collinear.