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.
Problem
Source: Sharygin Geometry Olympiad 2017 First Round P16 grades 9-11
Tags: geometry, concurrency, concurrent, tangent