APMO 2017 - Asian Pacific Mathematics Olympiad Question 2

Let $ABC$ be a triangle with $AB\lt AC$. Let $D$ be the second intersection of the internal bisector of $\angle BAC$ with the circumcircle of $ABC$. Let $Z$ be the intersection of the perpendicular bisector of $AC$ with the external bisector of $\angle BAC$. Prove that the midpoint of $AB$ lies on the circumcircle of $ADZ$.

Proposed by Equipo Nicaragua, Nicaragua.

Newly drawn configuration: