Isosceles Triangles Determined by a Planar Point Set

Abstract.

 It is proved that, for any ɛ>0 and n>n ₀(ɛ), every set of n points in the plane has at most triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2.136.) This easily implies the best currently known lower bound, , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi–Cs. Tóth and Tardos.