Question 169
On the side $BC$ of an acute triangle $ABC$ an arbitrary point $D$ is chosen. Let $O$ be the circumcenter of $\triangle ABC$, and $Z$ the point on this circle which is diametrically opposite to $A$. Let $X$, $Y$ be points on the segments $BO$, $CO$ such that:
$$
\angle BXD + \angle ABC = 180^{\circ} = \angle CYD + \angle ACB.
$$
Prove that the measure of the angle $\angle XZY$ does not depend on the point $D$.
(Khilko Danylo)
$$
\angle BXD + \angle ABC = 180^{\circ} = \angle CYD + \angle ACB.
$$
Prove that the measure of the angle $\angle XZY$ does not depend on the point $D$.
(Khilko Danylo)