  We consider a geometric percolation process partially motivated by recent
work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq
\mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T
\subseteq \mathbb{R}^2$ with its vertices in $\mathbb{Z}^2$ such that $T$
contains exactly four points of $\mathbb{Z}^2$ and exactly three points of $X$.
In this case, we add the missing lattice point of $T$ to $X$, and we repeat
until no such triangle exists. We study the limit sets $S$, the sets stable
under this process, including determining their possible densities and some of
their structure.
