KURSCHAK 2010 Problem 2

Consider a triangle $ABC$ , with the points $A_1$ , $A_2$ on side $BC$ , $B_1,B_2\in\overline{AC}$ , $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$ , $BA_1<BA_2$ , $CB_1<CB_2$ . Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$ . Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$ , let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$ . Prove that the lines $AA^*$ , $BB^*$ , $CC^*$ are concurrent.