Displacement of the excentral triangles

Question

The definition from the Wolfram MathWorld page: Excentral Triangle

Let $\triangle_1$ be the excentral triangle of a triangle $\triangle_0$, $\triangle_2$ the excentral triangle of $\triangle_1$, etc.

The Wolfram page states that the triangle $\triangle_n$ approaches an equilateral triangle as $n$ goes to infinity.

One may think that the position of the triangle $\triangle_n$ "stabilizes" as well.

However, this is not true for any non-equilateral triangle $\triangle_0$.
I found this fact while playing with the excentral triangles in GeoGebra.

Let $I_0$ be the incenter of $\triangle_0$, $I_1$ the incenter of $\triangle_1$, etc.
Then the vector $\overrightarrow{I_n, I_{n+1}}$ converges to some constant vector $\vec{I}$ such that $|\vec{I}| = 0$ if and only if $\triangle_0$ is equilateral.

Questions:

1. Is there a name for the vector $\vec{I}$?
2. How to find the size and direction of $\vec{I}$?
3. How is $\vec{I}$ related to the triangle centers of $\triangle_0$?