MK-MO 2021 Problem 3

Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$ . Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$ . The lines $AX$ and $BC$ intersect at $Y$ , while the lines $BX$ and $AD$ intersect at $Z$ . Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$ .