Hexyl Triangle

Given a triangle and the excentral triangle , define the -vertex of the hexyl triangle as the point in which the perpendicular to through the excenter meets the perpendicular to through the excenter , and similarly define and . Then is known as the hexyl triangle of , and forms a hexagon with parallel sides (Kimberling 1998 pp. 79 and 172).

The hexyl triangle has trilinear vertex matrix

(1)

where , , and (Kimberling 1998, p. 172).

It has side lengths

			(2)
			(3)
			(4)

and area

			(5)
			(6)
			(7)

where is the area of the reference triangle, is the circumradius, and is the inradius. It therefore has the same side lengths and area as the excentral triangle.

The Cevians triangles with Cevian points corresponding to Kimberling centers with , 20, 21, 27, 63, and 84 are perspective to the hexyl triangle. That anticevian triangles and antipedal triangles corresponding to Kimberling centers for , 9, 19, 40, 57, 63, 84, 610, 1712, and 2184 are also perspective to the hexyl triangle In fact, any point on the trilinear cubic

(8)

has an anticevian and antipedal triangle that are perspective with the hexyl triangle (P. Moses, pers. comm., Feb. 3, 2005).

The circumcircle of the hexyl triangle is the hexyl circle.

The triangle centroid of the hexyl triangle is the point with triangle center function

(9)

which is not a Kimberling center.

The following table gives the centers of the hexyl triangle in terms of the centers of the reference triangle for Kimberling centers with .

	center of hexyl triangle		center of reference triangle
	circumcenter		incenter
	orthocenter		Bevan point
	nine-point center		circumcenter
	triangle centroid of orthic triangle		triangle centroid of the antipedal triangle of
	orthocenter of orthic triangle		de Longchamps point
	symmedian point of orthic triangle		reflection of in
	Prasolov point		isogonal conjugate of
	Jerabek antipode		anticomplement of Feuerbach point
	Kiepert antipode		psi(incenter, symmedian point)
	Kiepert center		antipode of
	-of-medial triangle		lambda(incenter, triangle centroid)
	Jerabek center		antipode of

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