Question 173
Given an acute triangle $ABC$ and the center $O$ of its circumcircle. Let $O_1$ be a point on the perpendicular bisector of the segment $AB$ which lies on the opposite side of the line $AB$ from the point $O$. Denote by $\mathcal{K}$ the circle with center $O_1$ and radius $AO_1$. Let the lines $CA$ and $CB$ meet the circle $\mathcal{K}$ again at the points $A_1$ and $B_1$. Show: if the segments $A_1B$ and $AB_1$ meet on the circumcircle of the triangle $ABC$, then the quadrilateral $AO_1BO$ is cyclic.