For n points in the plane, there are at least N_1 = sqrt(n - 3/4) - 1/2 different distances. The minimum distance can occur only <=3n - 6 times, and the maximum distance can occur <=n times. Furthermore, no distance can occur as often as N_2 = 1/4 n(1 + sqrt(8n - 7))6 points in the plane can determine only isosceles triangles.