Compatible Triangulations

Is it true that every two sets of $n$ planar points in general position with the same number points on their convex hulls have compatible triangulations? Two triangulations are compatible if they have the same combinatorial structure, i.e., if their face lattices are isomorphic. For compatible triangulations $T_1$ and $T_2$ of point sets $S_1$ and $S_2$, there is a bijection $\phi$ between the points such that $ijk$ is a triangle of $T_1$ empty of points of $S_1$ iff $\phi(i) \phi(j) \phi(k)$ is a triangle of $T_2$ empty of points of $S_2$.