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Midpoint of bisector; prove that B, E, F, C cyclic

Source: Taiwan 2014 TST1, Problem 4

July 18, 2014
geometry proposedgeometry

Problem Statement

Let ABCABC be an acute triangle and let DD be the foot of the AA-bisector. Moreover, let MM be the midpoint of ADAD. The circle ω1\omega_1 with diameter ACAC meets BMBM at EE, while the circle ω2\omega_2 with diameter ABAB meets CMCM at FF. Assume that EE and FF lie inside ABCABC. Prove that BB, EE, FF, CC are concyclic.
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