Santa Claus decorates his Christmas tree with a decoration which has a shape of a regular $12$-sided polygon. Let the $12$-sided polygon be $A_1A_2A_3...A_{12}$. Suppose $I_1$, $I_2$ and $I_3$ are the incentres of $\vartriangle A_1A_2A_5$, $\vartriangle A_5A_7A_8$ and $\vartriangle A_8A_{11}A_1$ respectively. Prove that $I_1A_8$, $I_2A_1$ and $I_3A_5$ are concurrent.