First, some definitions: A mixtilinear incircle of a triangle is a circle that is tangent to two sides of the triangle and internally tangent to that triangle's circumcircle. There are three mixtilinear incircles for any nondegenerate triangle. The triangle connecting the centers of the three mixtilinear incircles is called the mixtilinear triangle of the first triangle.
My impression is that the mixtilinear triangle always seems to be smaller than the original triangle. So, assuming that (or not assuming it but only looking at triangles for which it is true), my question is as follows:
If one were to take the mixtilinear triangle of the mixtilinear triangle of the mixtilinear triangle of...of the original triangle, indefinitely, and it were to converge to a point, would that point have any geometric relationship to the original triangle? For example, might it be along the Euler line?
I'd like to note that, even assuming my above observation about the size of the mixtilinear incircle is true (and if it's not, I'd be curious to see a counterexample!), there's no way of procedurally determining which was the original triangle, so any property satisfied by that point with regard to the original triangle would also have to be satisfied with regard to any of the infinitely many mixtilinear and sub-mixtilinear triangles. In my view, that seems to make it substantially less likely that there is any nice property, but I'm still curious.
I came up with this problem myself- it's not from a textbook or anything like that, so I know of no ready-made solution just hiding out there. Also, I'm sorry for not adding more work, but I'm not generally very good at the inversive-geometry approach in which I encountered these triangles and from which I assume a proof would have to come.