Question 88
Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $D$ be a point on segment $BC$. The circle that passes through $B$ and $D$ and is tangent to $\Gamma$ and the circle that passes through $C$ and $D$ and is tangent to $\Gamma$, intersect at a point $E \neq D$. The line $DE$ intersects $\Gamma$ at two points, $X$ and $Y$. Prove that $|EX| = |EY|$.