SHARYGIN 2017 Problem 16

The tangents to the circumcircle of triangle $ABC$ at $A$ and $B$ meet at point $D$ . The circle passing through the projections of $D$ to $BC, CA, AB$ , meet $AB$ for the second time at point $C'$ . Points $A', B'$ are defined similarly. Prove that $AA', BB', CC'$ concur.