Given an Initial Triangle $A_1B_1 C_1$ construct the triangle $A_{n+1}B_{n+1}C_{n+1} $ from $A_n B_n C_n$, where $A_{n+1}$ is the $r_1$-th triangle center of $A_n B_n C_n$, $B_{n+1}$ is the $r_2$-th triangle center of $A_n B_n C_n$, $C_{n+1}$ is the $r_3$-th triangle center of $A_n B_n C_n$. Where the centers $r_1, r_2 , r_3$ are not collinear.
Like this construction I think this sequence will always converge except for a measure zero set, Proving this would be very difficult if not impossible (since the special case is hard enough) So I want to ask for a counter example, I am not very good at programming so doing a brute force or trial like this is not something I can do (I tried only 3 sets of centers and took me a lot of time)