Heptagonal Triangles and Their Companions

Paul Yiu
2009 Forum Geometricorum   unpublished
A heptagonal triangle is a non-isosceles triangle formed by three ver-tices 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
more » ... he 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.