  Erd\H{o}s and Fishburn studied the maximum number of points in the plane that
span $k$ distances and classified these configurations, as an inverse problem
of the Erd\H{o}s distinct distances problem. We consider the analogous problem
for triangles. Past work has obtained the optimal sets for one and two distinct
triangles in the plane. In this paper, we resolve a conjecture that at most six
points in the plane can span three distinct triangles, and obtain the hexagon
as the unique configuration that achieves this. We also provide evidence that
optimal sets cannot be on the square lattice in the general case.
