Converging Points

$\large {\left\{\begin{matrix}A_{n+1}=\alpha A_n+(1-\alpha)B_n \\ B_{n+1}=\alpha B_n+(1-\alpha)C_n \\ C_{n+1}=\alpha C_n+(1-\alpha)A_n\end{matrix}\right.}$

Let $A_1,B_1,C_1$ be three distinct non-collinear points on the coordinate plane.
Also $A_n,B_n,C_n$ satisfy the recurrence relation above ($0<\alpha<1$).

Then $\displaystyle\lim_{n\to\infty}A_n$ is the $\text{\_\_\_\_\_\_\_\_\_\_}$ of the triangle $A_1B_1C_1$.