Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. Prove that triangles ABC', AB'C, and A'BC are similar.
Problem
Source: Tournament of towns, Senior B-Level paper, Fall 2004
Tags: geometry, geometry solved