The incircle of triangle $A_1 A_2 A_3$ touches the sides $A_2 A_3$, $A_3 A_1$, and $A_1 A_2$ at $S_1$, $S_2$, and $S_3$, respectively. Let $O_1$, $O_2$, and $O_3$ be the incenters of triangles $A_1 S_2 S_3$, $A_2 S_3 S_1$, and $A_3 S_1 S_2$, respectively. Prove that $O_1 S_1$, $O_2 S_2$, $O_3 S_3$ are concurrent.
I have tried drawing a big diagram, as accurate as possible, as a start, and saw that $O_1$, $O_2$, and $O_3$ seem to touch the incircle of triangle $A_1A_2A_3$. However, I am not sure of how to prove this, or how this helps with proving the concurrency. Can I have a little hint as to how to prove this and how it helps answer the question? Much appreciated.