# Incremental structure of a delaunay triangulation

This would probably be considered a reference request, as I would imagine it has been studied extensively in earlier work. Say I have a collection of distinct points $X = \{x_1,\dots,x_n\}$ in the plane and let $T_n$ denote their Delaunay triangulation. Suppose I consider the set of all possible Delaunay triangulations of the $n+1$ points $X\cup x$ for all $x\in\mathbb{R^2}$. What is the maximum number of such triangulations? Here I'm considering two triangulations to be equivalent if their edge sets are identical.