BALKAN 2020 Problem 1

Let $A B C$ be an acute triangle with $A B=A C$, let $D$ be the midpoint of the side $A C$, and let $\gamma$ be the circumcircle of the triangle $A B D$. The tangent of $\gamma$ at $A$ crosses the line $B C$ at $E$. Let $O$ be the circumcentre of the triangle $A B E$. Prove that the midpoint of the segment $A O$ lies on $\gamma$.