KöMaL 2022 B5228

A parabola intersects side $AB$ of a triangle $ABC$ at interior points $C_1$ and $C_2$, side $BC$ at interior points $A_1$ and $A_2$, and it intersects side $CA$ at interior points $B_1$ and $B_2$. Prove that if $AC_1 = C_2B$ and $BA_1 = A_2C$ then $CB_1=B_2A$.