Let $P$ be a set of three noncollinear points in $\mathbb{R}^2$. Iteratively form the Delaunay triangulation $\cal T$ of $P$, and then augment $P$ by the circumcircle centers of all triangles in $\cal T$, repeating these two steps:
(1) ${\cal T} = \operatorname{DelTri}( P )$.
(2) $P = P \cup \{ x : x \; \textrm{is center of circumcircle of}\; \triangle \in {\cal T} \}$.
It appears that, after a few iterations, $P$ contains either an exactly zero-area triangle of three collinear points, or three nearly collinear points whose circumcircle center lies far outside the initial $P$. For example:
After $5$ iterations, subsets of $3$ nearly collinear points occur.
Starting with an equilateral triangle leads to collinearity in four steps:
Initial $P$ consists of corners of equilateral triangle.
Q. Is there any initial triangle that does not lead to a $3$-points-collinear or nearly collinear triangle?
By nearly collinear I mean that, as the number of iterations increases, the diameter of $P$ grows without bound: it grows because some triangles become very flat and their circumcenters are cast far away. Truly $3$-points-collinear would throw the circumcenter to $\infty$. I'm asking whether these conditions eventually necessarily occur for any initial triangle.
Added. Just a small amount of data, but $25$ random start triangles with coordinates uniform in $[-1,1]$ each reach a triangle of area smaller than $10^{-10}$ within $10$ iterations.
