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.

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