ELMO 2015 Problem 3

Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$ . Let $X$ be the reflection of $A$ across the point $B$ , and denote by $\gamma$ the circumcircle of triangle $BXC$ . Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$ . Prove that line $EX$ is tangent to the circle $\gamma$ .

Proposed by David Stoner