This is a development on my previous question Displacement of the excentral triangles
My idea is that a converging sequence of constructions may not be related to the known triangle centers, but rather to some other converging sequences.
And it looks like the limit vector from the question above is indeed related to another limit vector.
Let $\triangle_0$ be an arbitrary triangle with the incenter $I_0$.
Let us define the first sequence as:
$\triangle_{I_1}$ the intouch triangle of $\triangle_0$ with the incenter $I_1$,
$\triangle_{I_2}$ the intouch triangle of $\triangle_{I_1}$ with the incenter $I_2$,
...
$\triangle_{I_n}$ the intouch triangle of $\triangle_{I_{n-1}}$ with the incenter $I_{n}$,
and the second sequence as:
$\triangle_{E_1}$ the excentral triangle of $\triangle_0$ with the incenter $E_1$,
$\triangle_{E_2}$ the excentral triangle of $\triangle_{E_1}$ with the incenter $E_2$,
...
$\triangle_{E_n}$ the excentral triangle of $\triangle_{E_{n-1}}$ with the incenter $E_{n}$.
Then the direction of the vector $\overrightarrow{I_n, I_{n+1}}$ from the first sequence converges to a direction parallel to the limit vector of $\overrightarrow{E_n, E_{n+1}}$ from the second sequence as $n$ goes to infinity (the size of $\overrightarrow{I_n, I_{n+1}}$ obviously converges to $0$).
I have tested the construction in GeoGebra, and it seems to work with a good precision.
Each of the two red lines on the pictures is formed by two almost coinciding lines through the last two constructed vectors in each sequence:
Zoom in:
Does it make sense?
How can we prove (or disprove) the parallel property?
What is the limit point of the first $\overrightarrow{I_n, I_{n+1}}$ sequence?
Is there a catalogue of asymptotes and limit points of a triangle similar to the ETC?