External Similitude Center

In general, the external similitude center of two circles and with centers given in Cartesian coordinates is given by

(1)

In trilinear coordinates, the external center of similitude is given by , where

			(2)
			(3)
			(4)

The incircle and circumcircle of a triangle have two similitude centers, namely the internal similitude center Si and the external center of similitude Se. The external center of similitude of the circumcircle and incircle Se is the isogonal conjugate of the Nagel point of . It is Kimberling center and has equivalent triangle center functions

			(5)
			(6)
			(7)

The two points Si and Se share certain similar properties, but there seems to be no straightforward analogy between the two. For instance, the internal similitude center Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle taken pairwise, but the only comparable property of Se is more complicated: Se is the homothetic center of the tangential triangle and the reflection of the intangents triangle in the incenter of .

The following table summarized the external similitude centers for a number of named circles.

circle 1	circle 2	Kimberling	external similitude center
Adams' circle	Conway circle		incenter
Adams' circle	incircle		incenter
anticomplementary circle	Bevan circle		isogonal conjugate of
anticomplementary circle	Brocard circle		external similitude center( anticomplementary circle, Brocard circle)
anticomplementary circle	circumcircle		de Longchamps point
anticomplementary circle	cosine circle		point Castor II
anticomplementary circle	first Droz-Farny circle		orthocenter
anticomplementary circle	first Lemoine circle		external similitude center(anticomplementary circle, first Lemoine circle)
anticomplementary circle	first Johnson-Yff circle		intersection of and
anticomplementary circle	Gallatly circle		external similitude center(anticomplementary circle, Gallatly circle)
anticomplementary circle	incircle		crosspoint of the Gergonne point and Nagel point
anticomplementary circle	Moses circle		external similitude center(anticomplementary circle, Moses circle)
anticomplementary circle	nine-point circle		triangle centroid
anticomplementary circle	orthocentroidal circle		external similitude center(anticomplementary circle, orthocentroidal circle)
anticomplementary circle	polar circle		orthocenter
anticomplementary circle	Spieker circle		external similitude center(anticomplementary circle, Spieker center)
anticomplementary circle	Steiner circle		de Longchamps point
anticomplementary circle	tangential triangle		far-out point
Apollonius circle	Bevan circle		X(6)-Ceva conjugate of X(1)
Apollonius circle	Brocard circle		external similitude center(Brocard circle, Apollonius circle)
Apollonius circle	circumcircle		inverse-in-Brocard-circle of X(58)
Apollonius circle	cosine circle		external similitude center(2nd Lemoine circle, Apollonius circle)
Apollonius circle	excircles radical circle		external similitude center(Apollonius circle, excircles radical circle)
Apollonius circle	first Lemoine circle		external similitude center(1st Lemoine circle, Apollonius circle)
Apollonius circle	Gallatly circle		insimilicenter(Gallatly circle, Apollonius circle)
Apollonius circle	incircle		Apollonius point
Apollonius circle	nine-point circle		external similitude center(nine-point circle, Apollonius circle)
Bevan circle	Brocard circle		external similitude center(Bevan circle, Brocard circle)
Bevan circle	circumcircle		incenter
Bevan circle	cosine circle		external similitude center(Bevan circle, 2nd Lemoine circle)
Bevan circle	first Lemoine circle		external similitude center(Bevan circle, 1st Lemoine circle)
Bevan circle	Gallatly circle		external similitude center(Gallatly circle, Bevan circle)
Bevan circle	incircle		isogonal conjugate of X(9)
Bevan circle	Moses circle		external similitude center of excentral and Moses circles
Bevan circle	nine-point circle		external similitude center(Bevan circle, nine-point circle)
Bevan circle	Spieker circle		external similitude center(Bevan circle, Spieker circle)
Brocard circle	circumcircle		external similitude center(circumcircle, Brocard circle)
Brocard circle	cosine circle		insimilicenter(Brocard circle, 2nd Lemoine circle)
Brocard circle	first Lemoine circle		midpoint of Brocard diameter
Brocard circle	Gallatly circle		external similitude center(Gallatly circle, Brocard circle)
Brocard circle	incircle		external similitude center(incircle, Brocard circle)
Brocard circle	Moses circle		external similitude center(Moses circle, Brocard circle)
Brocard circle	nine-point circle		external similitude center(nine-point circle, Brocard circle)
Brocard circle	orthocentroidal circle		external similitude center(Brocard circle, orthocentroidal circle)
Brocard circle	second Brocard circle		external similitude center(circumcircle, 1st Lemoine circle)
Brocard circle	Spieker circle		external similitude center(Brocard circle, Spieker circle)
circumcircle	cosine circle		(X(3), X(6))-harmonic conjugate of X(371)
circumcircle	first Lemoine circle		external similitude center(circumcircle, 1st Lemoine circle)
circumcircle	Gallatly circle		inverse-in-Brocard-circle of X(1687)
circumcircle	half-Moses circle		symmedian point
circumcircle	incircle		external similitude center(circumcircle, incircle)
circumcircle	Moses circle		3rd power point
circumcircle	nine-point circle		orthocenter
circumcircle	orthocentroidal circle		external similitude center(circumcircle, orthocentroidal circle)
circumcircle	second Brocard circle		circumcenter
circumcircle	second Droz-Farny circle		circumcenter
circumcircle	second Johnson-Yff circle		Feuerbach point
circumcircle	sine-triple-angle circle		isogonal conjugate of X(847)
circumcircle	Steiner circle		de Longchamps point
circumcircle	tangential circle		Exeter point
Conway circle	excircles radical circle		Nagel point
Conway circle	incircle		incenter
cosine circle	first Lemoine circle		external similitude center(1st Lemoine circle, 2nd Lemoine circle)
cosine circle	Gallatly circle		inverse-in-Brocard-circle of X(1342)
cosine circle	incircle		(X(1), X(6))-harmonic conjugate of X(1124)
cosine circle	Lucas circles radical circle		circumcenter
cosine circle	Moses circle		external similitude center(Moses circle, 2nd Lemoine circles)
cosine circle	nine-point circle		inner Vecten point
cosine circle	orthocentroidal circle		external similitude center(cosine circle, orthocentroidal circle)
cosine circle	second Brocard circle		external similitude center(cosine circle, second Brocard circle)
cosine circle	Spieker circle		external similitude center(cosine circle, Spieker circle)
excircles radical circle	incircle		external similitude center(excircles radical circle, incircle)
excircles radical circle	nine-point circle		external similitude center(excircles radical circle, nine-point circle)
excircles radical circle	Spieker circle		Spieker center
extangents circle	nine-point circle		Clawson point
extangents circle	tangential circle		internal similitude center(circumcircle, incircle)
first Droz-Farny circle	polar circle		orthocenter
first Droz-Farny circle	second Droz-Farny circle		Euler infinity point
first Lemoine circle	Gallatly circle		(X(3), X(6))-harmonic conjugate of X(371)
first Lemoine circle	incircle		external similitude center(incircle, 1st Lemoine circle)
first Lemoine circle	Moses circle		external similitude center(Moses circle, 1st Lemoine circle)
first Lemoine circle	nine-point circle		external similitude center(first Lemoine circle, nine-point circle)
first Lemoine circle	orthocentroidal circle		external similitude center(first Lemoine circle, orthocentroidal circle)
first Lemoine circle	second Brocard circle		external similitude center(first Lemoine circle, second Brocard circle)
first Lemoine circle	Spieker circle		external similitude center(first Lemoine circle, Spieker circle)
first Johnson-Yff circle	incircle		orthocenter
first Johnson-Yff circle	nine-point circle		external similitude center(circumcircle, incircle)
first Johnson-Yff circle	second Johnson-Yff circle		orthocenter
Gallatly circle	half-Moses circle		Brocard midpoint
Gallatly circle	incircle		external similitude center(Gallatly circle, incircle)
Gallatly circle	Moses circle		Brocard midpoint
Gallatly circle	nine-point circle		external similitude center(Gallatly circle, nine-point circle)
Gallatly circle	orthocentroidal circle		external similitude center(Gallatly circle, orthocentroidal circle)
Gallatly circle	second Brocard circle		external similitude center(Gallatly circle, second Brocard circle)
Gallatly circle	Spieker circle		external similitude center(Gallatly circle, Spieker circle)
half-Moses circle	incircle		-isoconjugate of
half-Moses circle	Moses circle		Brocard midpoint
half-Moses circle	Spieker circle		external similitude center(half-Moses circle, Spieker circle)
incircle	inner Soddy circle		second Eppstein point
incircle	Moses circle		external similitude center(Moses circle, incircle)
incircle	nine-point circle		Feuerbach point
incircle	orthocentroidal circle		external similitude center(incircle, orthocentroidal circle)
incircle	outer Soddy circle		first Eppstein point
incircle	second Brocard circle		external similitude center(incircle, second Brocard circle)
incircle	second Johnson-Yff circle		orthocenter
incircle	sine-triple-angle circle		X(1)-Ceva conjugate of X(50)
incircle	Spieker circle		Nagel point
incircle	tangential mid-arc circle		third mid-arc point
inner Napoleon circle	outer Napoleon circle		triangle centroid
inner Soddy circle	Outer Soddy circle		Gergonne point
intangents circle	nine-point circle		intersection of and
Moses circle	nine-point circle		center of the Kiepert hyperbola
Moses circle	Spieker circle		external similitude center(Moses circle, Spieker circle)
nine-point circle	orthocentroidal circle		external similitude center(nine-point circle, orthocentroidal circle)
nine-point circle	second Brocard circle		external similitude center(nine-point circle, second Brocard circle)
nine-point circle	sine-triple-angle circle		focus of the Kiepert parabola
nine-point circle	Spieker circle		complementary conjugate of X(1)
nine-point circle	Steiner circle		nine-point center
nine-point circle	tangential circle		circumcenter
orthocentroidal circle	second Brocard circle		external similitude center(orthocentroidal circle, second Brocard circle)
orthocentroidal circle	Spieker circle		external similitude center(orthocentroidal circle, second Brocard circle)
second Brocard circle	second Droz-Farny circle		circumcenter
second Brocard circle	Spieker circle		external similitude center(second Brocard circle, Spieker circle)
sine-triple-angle circle	tangential circle		orthocorrespondent of