Converging Points

Geometry Level 2

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

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 \(A_1B_1C_1\).

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