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{An+1=αAn+(1−α)BnBn+1=αBn+(1−α)CnCn+1=αCn+(1−α)An \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.} ⎩⎪⎨⎪⎧An+1=αAn+(1−α)BnBn+1=αBn+(1−α)CnCn+1=αCn+(1−α)An
Let A1,B1,C1A_1,B_1,C_1A1,B1,C1 be three distinct non-collinear points on the coordinate plane. Also An,Bn,CnA_n,B_n,C_nAn,Bn,Cn satisfy the recurrence relation above (0<α<10<\alpha<10<α<1).
Then limn→∞An\displaystyle\lim_{n\to\infty}A_nn→∞limAn is the __________\text{\_\_\_\_\_\_\_\_\_\_} __________ of the triangle A1B1C1A_1B_1C_1A1B1C1.
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