Counting triangulations of balanced subdivisions of convex polygons

We compute the number of triangulations of a convex k-gon each of whose sides is subdivided by r−1 points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as k and/or r tend to infinity. We connect these results with the question of finding the planar set of n points in general position that has the minimum possible number of triangulations.