Geometry Problem 565: Triangle, Isogonal lines, Angles, Congruence, Concyclic Points

The figure shows a triangle ABC. Angles BAD
and CAE are congruent and AD = AE. Angles ABF and CBG are congruent and
BF = BG. Angles ACG and BCE are congruent and CE = CG. Prove
that the points D, E, G, and F are concyclic (lie on a same circle).