Question 197
Let $ABC$ be an acute-angled triangle with $AC \gt AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO$, $AC$ and $AB$ at $W$, $X$ and $Y$, respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \neq A$.
Prove that if $OW = OD$, then $DZ$ is tangent to the circle $AXY$.
Prove that if $OW = OD$, then $DZ$ is tangent to the circle $AXY$.