Let $n\ge 0$ and $A_0, A_1, A_2$ 3 points of the plane. We define recursively $A_{n+3}$ as the barycenter of the system $\{(A_n;2);(A_{n+1};1);(A_{n+2};1)\}$. Show that the sequence $(A_n)$ converges and find $A_{\infty}$.
My try: I define $z_n$ as the affix of $A_n$. This provide the relation: $4z_{n+3}-z_{n+2}-z_{n+1}-2z_n=0$. Solving this gives: $z_n=\lambda+\mu\rho_1^n+\nu\rho_2^n$. Since $ |\rho_{1,2}|<1$ the sequence converges and the limit will be $\lambda$ which I can find.
Is there an alternative prove specially with using geometry tools only?