SELECTION EXAMINATION 2019

Let $△AB\Gamma$ be a triangle with $AB>A\Gamma$. Let point $\Delta$ be on the side $AB$ such that $B\Delta =A\Gamma$. We draw the circle $\gamma$ passing through $\Delta$ and tangent to the side $A\Gamma$ at $A$. The circumcircle $\omega$ of the triangle $AB\Gamma$ meets circle $\gamma$ at $A$ and $E$. Prove that $E$ is the point of intersection of the perpendicular bisectors of the segments $B\Gamma$ and $A\Delta$.