Triangulations intersect nicely

We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.