Central Circle

A central circle is a circle whose equation in trilinear coordinates has the form

where is a triangle center and is a homogeneous function that is symmetric in the side lengths , , and (Kimberling 1998, p. 226).

The following table compares several familiar examples and includes the circumcircle for comparison, although it is excluded by the definition above. Here and are the circumradius and inradius of the reference triangle, is its semiperimeter, is its triangle centroid, is its orthocenter, and is its symmedian point. The reference triangle is gray and each circle is blue, with selected defining points and segments in orange. All drawings use Cartesian coordinates , , and and are independently scaled.

representative	circle	center	radius	defining property
	circumcircle	circumcenter		through , , and
	incircle	incenter		inside the reference triangle and in tangency with all three sides
	nine-point circle	nine-point center		through the side midpoints, altitude feet, and Euler points
	Brocard circle	midpoint of		diameter
	Bevan circle	Bevan point		circumcircle of the excentral triangle
	Spieker circle	Spieker center		incircle of the medial triangle
	orthocentroidal circle	midpoint of		diameter
	Conway circle	incenter		through the six extended-side points

The following table summarizes the triangle centers whose trilinears correspond to a circle with (for some appropriate value of ). In the table, indicated a circle function that is known but which does not appear among the list of Kimberling centers. Note also that the circumcircle is not actually a central circle, since its trilinears 0:0:0 are not those of a triangle center.

circle	Kimberling		Kimberling	center
Adams' circle	*			incenter
anticomplementary circle		third power point		orthocenter
Apollonius circle
Bevan circle		incenter		Bevan point
Brocard circle		triangle centroid		midpoint of the Brocard diameter
circumcircle	-	0		circumcenter
Conway circle				incenter
cosine circle				symmedian point
de Longchamps circle		third power point		de Longchamps point
Dou circle	*			eigencenter of the orthic triangle
Euler-Gergonne-Soddy circle	*		*
excircles radical circle		external similitude center of incircle and circumcircle		Spieker center
extangents circle	*		*
first Droz-Farny circle	*			orthocenter
first Johnson-Yff circle	*			first Johnson-Yff center
first Lemoine circle				midpoint of the Brocard diameter
Fuhrmann circle				Fuhrmann center
Gallatly circle				Brocard midpoint
GEOS circle	*		*
half-altitude circle	*		*
half-Moses circle	*			Brocard midpoint
hexyl circle	*			incenter
incentral circle			*
incircle				incenter
inner Napoleon circle		first isodynamic point		triangle centroid
inner Soddy circle	*			inner Soddy center
inner Vecten circle	*			complement of the inner Vecten point
intangents circle	*		*
Johnson triangle circumcircle		orthocenter		-Ceva conjugate of
Kenmotu circle		Kenmotu point		Cevapoint of and
Lester circle	*
Longuet-Higgins circle		Longuet-Higgins point		internal similitude center of Moses circle and incircle
Lucas central circle	*		*
Lucas circles radical circle		triangle centroid
Lucas inner circle		incenter	*
Macbeath circle	*		*
Mandart circle				circumcenter of the extouch triangle
McCay circles radical circle	*		*
mixtilinear circle	*		*
mixtilinear incircles radical circle		mittenpunkt		midpoint of
Morley's circle	*			first Morley center
Moses circle		isotomic conjugate of		Brocard midpoint
Moses-Longuet-Higgins circle	*			-Ceva conjugate of
Neuberg circles radical circle	*			-Ceva conjugate of
nine-point circle		circumcenter		nine-point center
orthocentroidal circle		circumcenter		midpoint of
orthoptic circle of the Steiner inellipse		circumcenter		triangle centroid
outer Napoleon circle		second isodynamic point		triangle centroid
outer Soddy circle	*			outer Soddy center
outer Vecten circle	*			complement of the outer Vecten point
Parry circle
polar circle		circumcenter		orthocenter
reflection circle		-Ceva conjugate of		-Ceva conjugate of
second Brocard circle		symmedian point		circumcenter
second Droz-Farny circle		symmedian point		circumcenter
second Johnson-Yff circle	*			second Johnson-Yff center
second Steiner circle	*		*
sine-triple-angle circle	*
Spieker circle	*			Spieker center
Stammler circle		symmedian point		circumcenter
Stammler circles radical circle				nine-point center
Steiner circle	*			nine-point center
Stevanović circle
symmedial circle			*
tangential circle		circumcenter		circumcenter of the tangential triangle
tangential mid-arc circle			*
Taylor circle				Taylor center
third Lemoine circle	*		*
van Lamoen circle	*
Yff central circle	*		*
Yff contact circle	*		*
Yiu circle	*		*

The following table summarizes circles sorted by center and indicates concentric circles.

Kimberling	center	circles
	incenter	Adams' circle, Conway circle, hexyl circle, incircle
	triangle centroid	inner Napoleon circle, orthoptic circle of the Steiner inellipse, outer Napoleon circle
	circumcenter	circumcircle, second Brocard circle, second Droz-Farny circle, Stammler circle
	orthocenter	anticomplementary circle, Johnson triangle circumcircle, polar circle, first Droz-Farny circle
	nine-point center	nine-point circle, Stammler circles radical circle, Steiner circle
	symmedian point	cosine circle
	Spieker center	excircles radical circle, Spieker circle
	de Longchamps point	de Longchamps circle
	circumcenter of the tangential triangle	tangential circle
	Brocard midpoint	Gallatly circle, half-Moses circle, Moses circle
	Bevan point	Bevan circle
	center of the sine-triple-angle circle	sine-triple-angle circle
	eigencenter of orthic triangle	Dou circle
	isoperimetric point	outer Soddy circle
	equal detour point	inner Soddy circle
	midpoint of the Brocard diameter	Brocard circle, first Lemoine circle
	-Ceva conjugate of	Neuberg circles radical circle
	-Ceva conjugate of	reflection circle
	center of the Parry circle	Parry circle
	first Morley center	Morley's circle
	midpoint of and	orthocentroidal circle
		outer Vecten circle
		inner Vecten circle
		Stevanović circle
	Longuet-Higgins point	Longuet-Higgins circle
		Apollonius circle
		mixtilinear incircles radical circle
		Lester circle
		Lucas circles radical circle
		van Lamoen circle
		Mandart circle
		first Johnson-Yff circle
		second Johnson-Yff circle