2011 JBMO Shortlist G1

Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$ and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic.

posted in aops here

my solution