Classification of isosceles eight-point sets in three-dimensional Euclidean space

A subset X in k -dimensional Euclidean space R k that contains n points (elements) is called a P( n )-set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that the P(8)-set in R 3 is uniquely determined to the known example in Kelly’s paper [L.M. Kelly, Elementary problems and solutions. Isosceles n -points, Amer. Math. Monthly 54 (1947) 227–229].