Let V be a point set in a Euclidean space. We prove that if ∣V∣ ⩾ 5 and all triangles, each spanned by a triple of V, have the same area > 0 then V forms the vertex set of a regular simplex. Further, if ∣V∣ is large and the triangles have r ⩾ 2 different areas then the number of different distances between the pairs of V is at most 2r3(2r + 1).
