The horizon of the heptagonal triangle

Question

I have discovered this interesting property of the heptagonal triangle while making constructions in GeoGebra.

Let $\triangle_0$ be a heptagonal triangle with the orthocenter $H_0$,
$\triangle_1$ the orthic triangle of $\triangle_0$ with the orthocenter $H_1$,
...
$\triangle_n$ the orthic triangle of $\triangle_{n-1}$ with the orthocenter $H_n$.

Let $H$ be the homothetic center of the triangles $\triangle_0$, $\triangle_2$, $\triangle_4$, ..., $\triangle_{2k}$.
Obviously, $H$ is also the homothetic center of the triangles $\triangle_1$, $\triangle_3$, $\triangle_5$, ..., $\triangle_{2k + 1}$.

It appears all the orthocenters $H_0$, $H_1$, ..., $H_n$ lie on a straight line passing through $H$.
Furthermore, the sequence $H_0$, $H_1$, $H_2$, ... converges to $H$.
Let us name the line the horizon of $\triangle_0$.

Questions:

1. Is there a simple way to prove the existence of the horizon of the heptagonal triangle?
2. Are there other interesting properties of the line?
3. Are there other triangles with the same or similar property?