I am relatively new to this topic, so this question may be easy/naive to some experts. Here goes..

I have a finite set of points $S\subset\mathbb R^2$ (you may increase the dimension of the ambient space if you feel challenged). And I take a coherent triangulation $\cal T$ (see [GKZ] for definition) of $S$ (sometimes also called "upper convex-hull" or regular triangulation). Suppose now that we include more points to $S$, i.e. we consider a a finite set $S\cup U$, such that $U$ belongs in the convex hull of $S$. Take coherent triangulations of all the points in $T\cap (S\cup U)$ for each triangle $T \in \cal T$.

My question: Is the union of the triangles in each of these triangulations a coherent triangulation of $S\cup U$?

[GKZ]: Gelfand I.M., Kapranov M.M and Zelevinsky A.V. Discriminants, Resultants and Multidimensional Resultants. Chapter §7.1