Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$ are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.