BALKAN-SL 2007 G1

Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$ . The tangents from $A$ touch $\omega$ at points $B$ , and $C$ . Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$ . Denote by $X$ the orthogonal projection of $B$ onto $CD$ , by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$ . Prove that $ZA$ and $ZC$ are perpendicular to each other.