Convergence of the sequence of the orthocenters, incenters and centroids of a triangle.

Now asked on MO here.

Given a triangle $A_1B_1C_1$ from the triangle $A_nB_nC_n$ construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is the incenter of $A_nB_nC_n$ and $C_{n+1}$ is the centroid of $A_nB_nC_n$, My question is If this process is repeated indefinitely would the sequences $\{A_n\}, \ \{B_n\}, \ \{C_n\} $ converge ?

There are only four possible scenarios:

1. The points will converge to a point.
2. The points will eventually stuck on a finite loop.
3. The points will completely diverge.
4. The points will eventually make an isosceles triangle (or equilateral triangle) which will end the sequence.

All of these scenarios would depend on the starting tingle triangle. So given $\angle A_1, \ \angle B_1, \ \angle C_1, $ how to determine which of the four scenario would happen to the sequences $\{A_n\}, \ \{B_n\}, \ \{C_n\} $? , I tried to see if the sequence of area would converge or not to give better information about the sequence but the numbers seem to be chaotic. So another question is Whether the sequence of the area of the triangle $A_nB_nC_n$ will converge.

I tried to draw some random triangles and see if the point will converge

It seems that that the points will converge in most cases to a point but I am not sure how to prove or disprove that.