SHARYGIN 2012 Problem 11

Given triangle $ABC$ and point $P$ . Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$ . A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$ . Points $A_{1}, B_{1}$ are defined similarly. Prove that a) lines $AA_{1}, BB_{1}, CC_{1}$ concur; b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.