I asked this question on MSE [here][1].


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Given a non-regular pentagon $A_1B_1C_1D_1E_1$, from the pentagon  $A_nB_nC_nD_nE_n$ construct the pentagon  $A_{n+1}B_{n+1}C_{n+1}D_{n+1}E_{n+1}$ as follows:

 - $A_{n+1}$ is the intersection between the angle bisector of $\angle
   C_n $ and $\angle D_n$. 
 - $B_{n+1}$ is the intersection between the angle bisector of $\angle D_n$ and $\angle E_n$.
 - $C_{n+1}$ is the intersection between the angle bisector of $\angle E_n$ and $\angle A_n$.
 -  $D_{n+1}$ is the intersection between the angle bisector of $\angle A_n$ and $\angle B_n$.
 - $E_{n+1}$ is the intersection between the angle bisector of $\angle B_n$ and $\angle C_n$.

 **(the two opposite angles)** The reason that I chose this construction is that the point $A_{n+1}$ is the only point that doesn't  depend on the position of $A_n$. I am allowing self-intersecting polygons in this constructions.  

My question is if this process is repeated indefinitely would the sequences ${A_n}$
, ${B_n}$
, ${C_n}$ , $D_n $ and $E_n$
 converge ?

There are only four possible scenarios:

 

 1. The points will converge.
 2. The points will eventually trapped  on a loop.
 3. The points will diverge completely.
 4. Two or more points coincide, ending the sequence.




 

 
I tried to draw the first few pentagons to  see if the point will converge or not.

[![enter image description here][2]][2]
[![enter image description here][3]][3]

Here is the first 50 pentagons in a zoom-in animation: 
[![enter image description here][4]][4]

I conjecture that the sequences converge to a single point for all non-regular pentagons. If convergence occurs, how can we determine the limit point based on the initial pentagon?

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[Here][5] is a Geogebra file that have the first 50 pentagons. 


  [1]: https://math.stackexchange.com/questions/4937560/does-the-sequence-formed-by-intersecting-angle-bisector-in-a-pentagon-converges
  [2]: https://i.sstatic.net/8M3RUVFT.png
  [3]: https://i.sstatic.net/2fbkfWiM.png
  [4]: https://i.sstatic.net/Ddnk9Px4.gif
  [5]: https://www.geogebra.org/classic/tywbvnyg