Isosceles Triangles Determined by a Planar Point Set

 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. Received: February, 2002 Final version received: September 15, 2002 RID="*" ID="*" Supported by NSF grant CCR-00-86013, PSC-CUNY Research Award 63382-00-32, and OTKA-T-032452 RID="†" ID="†" Supported by OTKA-T-030059 and AKP 2000-78-21