In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $ A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.
Problem
Source: Sharygin 2006 IX- X CR 11
Tags: concurrency, concurrent, Circumcenter, geometry, circumcircle