Heptagonal Triangles and Their Companions

A heptagonal triangle is a non-isosceles triangle formed by three vertices of a regular heptagon. Its angles are π 7 , 2π 7 and 4π 7 . As such, there is a unique choice of a companion heptagonal triangle formed by three of the remaining four vertices. Given a heptagonal triangle, we display a number of interesting companion pairs of heptagonal triangles on its nine-point circle and Brocard circle. Among other results on the geometry of the heptagonal triangle, we prove that the circumcenter and the Fermat points of a heptagonal triangle form an equilateral triangle. The proof is an interesting application of Lester’s theorem that the Fermat points, the circumcenter and the nine-point center of a triangle are concyclic. 1. The heptagonal triangle T and its companion A heptagonal triangle T is one with angles π7 , 2π 7 and 4π 7 . Its vertices are three vertices of a regular heptagon inscribed in its circumcircle. Among the remaining four vertices of the heptagon, there is a unique choice of three which form another (congruent) heptagonal triangle T′. We call this the companion of T, and the seventh vertex of the regular heptagon the residual vertex of T and T′ (see Figure 1). In this paper we work with complex number coordinates, and take the unit circle