USAMO 2011 Problem 3

In hexagon $ABCDEF$ , which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$ , $\angle C=3\angle F$ , and $\angle E=3\angle B$ . Furthermore $AB=DE$ , $BC=EF$ , and $CD=FA$ . Prove that diagonals $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ are concurrent.