We examine the number of triangulations that any set of n points in the plane
must have, and prove that (i) any set of n points has at least 0.00037*2.2n
triangulations, (ii) any set with three extreme points and n interior points
has at least 0.112*2.569n triangulation, and (iii) any set with n interior
points has at least 0.238 * 2.38n triangulation. The best previously known
lower bound for the number of triangulations for n points in the plane is
0.0822 * 2.0129n. We also give a method of automatically extending known
bounds for small point sets to general lower bounds.