Paul Yiu, Heptagonal triangles and their companions,
Forum Geometricorum, 9 (2009) 125--148.
Abstract: A heptagonal triangle is a non-isosceles triangle formed
by three vertices of a regular heptagon. Its angles are Pi/7, 2Pi/7, 4Pi/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.