Radical Line

The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles. By the chordal theorem, it is perpendicular to the line of centers (Dörrie 1965).

Let the circles have radii and and their centers be separated by a distance . If the circles intersect in two points, then the radical line is the line passing through the points of intersection. If not, choose two auxiliary circles of different radii with a common center off the line of centers of the original circles, each cutting both originals twice. For each auxiliary circle, draw the two lines through its pairs of points of intersection with the original circles. Their intersection is a radical center. The two radical centers are distinct, and the line joining them is the radical line.

Given two circles with equations in trilinear coordinates

(1)

their radical line has equation

(2)

(Kimberling 1998, p. 224).

Let and be the circle centers, and orient their line from toward . The signed distances from these centers to the radical line are

			(3)
			(4)

respectively, where

(5)

The radical line of any two polar circles is the altitude from the third vertex.

The following table gives the radical lines of pairs of circles that correspond to Kimberling centers

circle 1	circle 2	line	line name
anticomplementary circle	Bevan circle
anticomplementary circle	Brocard circle
anticomplementary circle	circumcircle		de Longchamps line
anticomplementary circle	Conway circle
anticomplementary circle	de Longchamps circle		de Longchamps line
anticomplementary circle	Fuhrmann circle
anticomplementary circle	Stammler circle
Apollonius circle	circumcircle
Apollonius circle	excircles radical circle		antiorthic axis
Apollonius circle	nine-point circle		antiorthic axis
Bevan circle	circumcircle		antiorthic axis
Bevan circle	Conway circle
Bevan circle	de Longchamps circle
Bevan circle	Dou circle
Bevan circle	excircles radical circle		Gergonne line
Bevan circle	extangents circle
Bevan circle	Fuhrmann circle
Bevan circle	polar circle		Gergonne line
Brocard circle	circumcircle		Lemoine axis
Brocard circle	cosine circle
Brocard circle	de Longchamps circle
Brocard circle	Lucas circles radical circle		Lemoine axis
Brocard circle	Lucas inner circle		Lemoine axis
Brocard circle	nine-point circle
Brocard circle	orthocentroidal circle
Brocard circle	orthoptic circle of the Steiner inellipse
Brocard circle	polar circle
Brocard circle	second Brocard circle
circumcircle	Conway circle
circumcircle	cosine circle
circumcircle	de Longchamps circle		de Longchamps line
circumcircle	Dou circle
circumcircle	excircles radical circle
circumcircle	first Lemoine circle
circumcircle	Fuhrmann circle
circumcircle	Gallatly circle
circumcircle	incentral circle
circumcircle	incircle
circumcircle	inner Napoleon triangle
circumcircle	Lucas circles radical circle		Lemoine axis
circumcircle	Lucas inner circle		Lemoine axis
circumcircle	Mandart circle
circumcircle	Moses circle
circumcircle	nine-point circle		orthic axis
circumcircle	orthocentroidal circle		orthic axis
circumcircle	orthoptic circle of the Steiner inellipse		orthic axis
circumcircle	outer Napoleon circle
circumcircle	Parry circle
circumcircle	polar circle		orthic axis
circumcircle	reflection circle
circumcircle	Stammler circles radical circle
circumcircle	Stevanovic circle
circumcircle	symmedial circle
circumcircle	tangential circle		orthic axis
circumcircle	Taylor circle
Conway circle	de Longchamps circle
cosine circle	first Lemoine circle
de Longchamps circle	incircle
de Longchamps circle	Neuberg circles radical circle
de Longchamps circle	nine-point circle
de Longchamps circle	polar circle
de Longchamps circle	second Steiner circle
de Longchamps circle	Yff contact circle		Nagel line
Dou circle	polar circle
Euler-Gergonne-Soddy circle	GEOS circle		Soddy line
excircles radical circle	Mandart circle
excircles radical circle	nine-point circle		antiorthic axis
excircles radical circle	polar circle		Gergonne line
first Droz-Farny circle	nine-point circle
first Droz-Farny circle	second Droz-Farny circle
first Lemoine circle	nine-point circle
Fuhrmann circle	polar circle
half-altitude circle	nine-point circle
half-altitude circle	polar circle
incircle	inner Soddy circle		Gergonne line
incircle	mixtilinear incircles radical circle
incircle	nine-point circle
incircle	outer Soddy circle		Gergonne line
inner Napoleon circle	outer Napoleon circle
inner Soddy circle	outer Soddy circle		Gergonne line
inner Vecten circle	outer Vecten circle
Lester circle	orthocentroidal circle
Lucas circles radical circle	Lucas inner circle		Lemoine axis
Mandart circle	nine-point circle
mixtilinear incircles radical circle	polar circle
Moses circle	nine-point circle
nine-point circle	orthocentroidal circle		orthic axis
nine-point circle	orthoptic circle of the Steiner inellipse		orthic axis
nine-point circle	polar circle		orthic axis
nine-point circle	Spieker circle
nine-point circle	tangential circle		orthic axis
nine-point circle	Taylor circle
nine-point circle	first Droz-Farny circle
orthocentroidal circle	orthoptic circle of the Steiner inellipse		orthic axis
orthocentroidal circle	polar circle		orthic axis
orthocentroidal circle	reflection circle
orthocentroidal circle	tangential circle		orthic axis
orthoptic circle of the Steiner inellipse	polar circle		orthic axis
orthoptic circle of the Steiner inellipse	tangential circle		orthic axis
polar circle	second Droz-Farny circle
polar circle	tangential circle		orthic axis
polar circle	Taylor circle
Stammler circle	Stammler circles radical circle
Stammler circles radical circle	first Droz-Farny circle
Stammler circles radical circle	second Droz-Farny circle