Let ABC be an acute triangle and D a point on the side BC such that ∠BAD=∠DAC. The circumcircles of the triangles ABD and ACD intersect the segments AC and AB at E and F, respectively. The internal bisectors of ∠BDF and ∠CDE intersect the sides AB and AC at P and Q, respectively. Points X and Y are chosen on the side BC such that PX is parallel to AC and QY is parallel to AB. Finally, let Z be the point of intersection of BE and CF. Prove that ZX=ZY. geometryequal segmentsequal angles