Anticomplementary Triangle

The anticomplementary triangle is the triangle which has a given triangle as its medial triangle. It is therefore the anticevian triangle with respect to the triangle centroid (Kimberling 1998, p. 156), and is in perspective with at .

It is the polar triangle of the Steiner circumellipse.

Its trilinear vertex matrix is

(1)

or

(2)

The sides of the anticomplementary triangle are 's exmedians and the vertices are the exmedian points of .

The circumcircle of the anticomplementary triangle is the anticomplementary circle.

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

	center of anticomplementary triangle		center of reference triangle
	incenter		Nagel point
	triangle centroid		triangle centroid
	circumcenter		orthocenter
	orthocenter		de Longchamps point
	nine-point center		circumcenter
	symmedian point		symmedian point of the anticomplementary triangle
	Gergonne point		anticomplement of
	Nagel point		anticomplement of Nagel point
	mittenpunkt		Gergonne point
	Spieker center		incenter
	Feuerbach point		anticomplement of Feuerbach point
	(,)-harmonic conjugate of		internal similitude center(circumcircle, AC-incircle)
	first Fermat point		anticomplement of
	second Fermat point		anticomplement of
	first isodynamic point		anticomplement of
	second isodynamic point		anticomplement of
	first Napoleon point		anticomplement of
	second Napoleon point		anticomplement of
	Schiffler point		anticomplement of
	Euler infinity point		Euler infinity point
	homothetic center of orthic and tangential triangles		anticomplement of
	third power point		isotomic conjugate of
	crosspoint of incenter and triangle centroid		isotomic conjugate of incenter
	Brocard midpoint		third Brocard point
	Bevan point		Longuet-Higgins point
	-line conjugate of		isotomic conjugate of
	triangle centroid of orthic triangle		triangle centroid of dual triangle of
	Kosnita point		anticomplementary conjugate of
	isogonal conjugate of		isotomic conjugate of
	isogonal conjugate of		anticomplementary conjugate of
	isogonal conjugate of		anticomplement of
	isogonal conjugate of		anticomplement of
	symmedian point of the anticomplementary triangle		-Ceva conjugate of
			reflection of in
	isotomic conjugate of incenter		-Ceva conjugate of
	third Brocard point		-Ceva conjugate of
	cevapoint of incenter and symmedian point		anticomplementary conjugate of
	cevapoint of triangle centroid and symmedian point		anticomplementary conjugate of
	cevapoint of incenter and triangle centroid		first Hatzipolakis parallelian point
	Tarry point		Tarry point of anticomplementary triangle
	Steiner point		Steiner point of anticomplementary triangle
	anticomplement of Feuerbach point		reflection of in

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 central triangle each have sides of length , , and . Similarly, each of these triangles, including , have area

(6)

where is the triangle area of .