Characterizing optimal point sets determining one distinct triangle

In this paper we determine the maximum number of points in ${\mathbb{R}}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t=1$. We denote this quantity by $F_d(t)$. It was known from the work of Epstein et al. that $F_2(1)=4$. Here we show somewhat surprisingly that $F_3(1)=4$ and $F_d(1)=d+1$, whenever $d\ge 3$, and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erd\H{o}s and Fishburn.