SHARYGIN 2014 Problem 13

Let $AC$ be a fixed chord of a circle $\omega$ with center $O$ . Point $B$ moves along the arc $AC$ . A fixed point $P$ lies on $AC$ . The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$ , the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$ . Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.