Medial Triangle

The medial triangle is the triangle formed by joining the midpoints of the sides of a triangle . The medial triangle is sometimes also called the auxiliary triangle (Dixon 1991).

The medial triangle is the Cevian triangle of the triangle centroid and the pedal triangle of the circumcenter (Kimberling 1998, p. 155). It is also the cyclocevian triangle of the orthocenter .

The medial triangle is the polar triangle of the Steiner inellipse.

Its trilinear vertex matrix is

(1)

or

(2)

The medial triangle of a triangle is similar to and its side lengths are

			(3)
			(4)
			(5)

This follows immediately by inspecting the construction of the medial triangle and noting that the three vertex triangles and medial triangle each have sides of length , , and . Similarly, each of these triangles, including , have area

(6)

where is the triangle area of .

The incircle of the medial triangle is called the Spieker circle, and its incenter is called the Spieker center. The circumcircle of the medial triangle is the nine-point circle.

Given a reference triangle , let either the internal or external angle bisectors of and cut the side (or extended side) of the medial triangle at finite points and . Then is perpendicular to the angle bisector of and is perpendicular to the angle bisector of . Similarly, by taking pairs of angle bisectors in turn, perpendiculars can be dropped from and to their respective intersections with the other sides of the medial triangle (Carding 2006, pp. 129-131; F. M. Jackson, pers. comm., Aug. 5, 2006 and June 12, 2008).

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

	center of medial triangle		center of reference triangle
	incenter		Spieker center
	triangle centroid		triangle centroid
	circumcenter		nine-point center
	orthocenter		circumcenter
	nine-point center		midpoint of and
	symmedian point		complement of symmedian point
	Gergonne point		mittenpunkt
	Nagel point		incenter
	mittenpunkt		complement of mittenpunkt
	Spieker center		complement of
	Feuerbach point		complement of
	first Fermat point		complement of
	second Fermat point		complement of
	first isodynamic point		complement of
	second isodynamic point		complement of
	first Napoleon point		complement of
	second Napoleon point		complement of
	de Longchamps point		orthocenter
	Schiffler point		complement of Schiffler point
	Exeter point		complement of
	far-out point		complement of
	homothetic center of orthic and tangential triangles		complementary conjugate of
	Cevapoint of orthocenter and Clawson center		complement of
	Euler infinity point		Euler infinity point
	second power point		complementary conjugate of
	third power point		complement of
	crosspoint of and		isogonal conjugate of
	Bevan point		midpoint of and
	orthocenter of orthic triangle		isogonal conjugate of
	Kosnita point		isogonal conjugate of
	internal similitude center of circumcircle and incircle		complementary conjugate of
	external similitude center of circumcircle and incircle		complementary conjugate of
	isogonal conjugate of		complement of
	isogonal conjugate of		complement of
	isogonal conjugate of		-Ceva conjugate of
	isogonal conjugate of		complementary conjugate of
	orthocenter of the intouch triangle		intersection of lines and
	isogonal conjugate of		-Ceva conjugate of
	Prasolov point		isogonal conjugate of
	symmedian point of the anticomplementary triangle		symmedian point
	isogonal conjugate of		inverse-in-incircle of
			Jerabek antipode
	isotomic conjugate of incenter		crosspoint of incenter and triangle centroid
	third Brocard point		Brocard midpoint
	isogonal conjugate of		isogonal conjugate of
	reflection of incenter in Feuerbach point		-Ceva conjugate of
	Cevapoint of incenter and symmedian point		isogonal conjugate of
	isotomic conjugate of		isogonal conjugate of
	Cevapoint of incenter and triangle centroid		isogonal conjugate of
	Cevapoint of incenter and Clawson point		isogonal conjugate of
	Cevapoint of triangle centroid and circumcenter		-Ceva conjugate of
	Tarry point		Kiepert antipode
	Steiner point		center of Kiepert hyperbola
	anticomplement of Feuerbach point		Feuerbach point