China Southeastern Mathematical Olympiad

Let $A{A}_0$, $B{B}_0$ and $C{C}_0$ be angular bisectors of $△ABC$. Let $A_0{A}_1\parallel B{B}_0$ and $A_0{A}_2\parallel C{C}_0$, where $A_1$ and $A_2$ lie on $AC$ and $AB$, respectively, and let line $A_1{A}_2$ intersect $BC$ at $A_3$. The points $B_3$ and $C_3$ are obtained similarly. Prove that points $A_3$, $B_3$, $C_3$ are collinear. (posed by Tao Pingsheng)