Question 116
Let $ABC$ be an acute-angled triangle with $AC \gt AB$, and let $O$ be its circumcenter. Let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ meets the lines $AO$, $AC$ and $AB$ at $W$, $X$ and $Y$, respectively. The circumcircles of triangles $AXY$ and $ABC$ meet again at $Z \neq A$.
Suppose $D \neq W$ and $OW = OD$. Prove that $DZ$ is tangent to the circumcircle of triangle $AXY$.
Suppose $D \neq W$ and $OW = OD$. Prove that $DZ$ is tangent to the circumcircle of triangle $AXY$.