  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 points in general position that has the minimum
possible number of triangulations - a well-known open problem from
computational geometry.
