Internal Similitude Center

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

(1)

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

			(2)
			(3)
			(4)

The incircle and circumcircle of a triangle have two similitude centers, namely the internal center of similitude Si and the external similitude center Se. The internal center of similitude of these two circles Si is the isogonal conjugate of the Gergonne 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, Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle taken pairwise, but the only comparable property of the external similitude center 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 summarizes the internal similitude centers for a number of named circles.

circle 1	circle 2	Kimberling	internal similitude center
Adams circle	Conway circle		incenter
Adams circle	incircle		incenter
anticomplementary circle	Bevan circle		Spieker center
anticomplementary circle	Brocard circle		insimilicenter(anticomplementary circle, Brocard circle)
anticomplementary circle	circumcircle		triangle centroid
anticomplementary circle	cosine circle		point Castor I
anticomplementary circle	first Droz-Farny circle		orthocenter
anticomplementary circle	first Lemoine circle		insimilicenter(anticomplementary circle, first Lemoine circle)
anticomplementary circle	Gallatly circle		insimilicenter(anticomplementary circle, Gallatly circle)
anticomplementary circle	incircle		intersection of lines and
anticomplementary circle	Moses circle		insimilicenter(anticomplementary circle, Moses circle)
anticomplementary circle	orthocentroidal circle		insimilicenter(anticomplementary circle, orthocentroidal circle)
anticomplementary circle	polar circle		orthocenter
anticomplementary circle	second Johnson-Yff circle		crosspoint of the Nagel point and Gergonne point
anticomplementary circle	Spieker circle		insimilicenter(anticomplementary circle, Spieker circle)
Apollonius circle	Bevan circle		insimilicenter(Bevan circle, Apollonius circle)
Apollonius circle	Brocard circle		insimilicenter(Brocard circle, Apollonius circle)
Apollonius circle	circumcircle		orthoharmonic of X(58)
Apollonius circle	cosine circle		insimilicenter(2nd Lemoine circle, Apollonius circle)
Apollonius circle	excircles radical circle		insimilicenter(Apollonius circle, Brocard circle)
Apollonius circle	first Lemoine circle		insimilicenter(first Lemoine circle, excircles radical circle)
Apollonius circle	Gallatly circle		exsimilicenter(Gallatly circle, Apollonius circle)
Apollonius circle	incircle		insimilicenter(incircle, Apollonius circle)
Apollonius circle	nine-point circle		Spieker center
Bevan circle	Brocard circle		insimilicenter(Bevan circle, Brocard circle)
Bevan circle	circumcircle		centroid of the excentral triangle
Bevan circle	cosine circle		insimilicenter(Bevan circle, 2nd Lemoine circle)
Bevan circle	first Lemoine circle		insimilicenter(Bevan circle, 1st Lemoine circle)
Bevan circle	Gallatly circle		insimilicenter(Gallatly circle, Bevan circle)
Bevan circle	incircle		insimilicenter(Bevan circle, incircle)
Bevan circle	Moses circle		insimilicenter of excentral and Moses circles
Bevan circle	nine-point circle		insimilicenter(Bevan circle, nine-point circle)
Bevan circle	second Brocard circle		insimilicenter(Bevan circle, second Brocard circle)
Bevan circle	Spieker circle		mittenpunkt
Brocard circle	circumcircle		insimilicenter(circumcircle, Brocard circle)
Brocard circle	cosine circle		exsimilicenter(Brocard circle, 2nd Lemoine circle)
Brocard circle	first Lemoine circle		midpoint of Brocard diameter
Brocard circle	Gallatly circle		insimilicenter(Gallatly circle, Brocard circle)
Brocard circle	incircle		insimilicenter(incircle, Brocard circle)
Brocard circle	Moses circle		insimilicenter(Moses circle, Brocard circle)
Brocard circle	nine-point circle		insimilicenter(nine-point circle, Brocard circle)
Brocard circle	orthocentroidal circle		insimilicenter(Brocard circle, orthocentroidal circle)
Brocard circle	second Brocard circle		insimilicenter(circumcircle, 1st Lemoine circle)
Brocard circle	Spieker circle		insimilicenter(Brocard circle, Spieker circle)
circumcircle	cosine circle		Kenmotu point (congruent squares point)
circumcircle	excircles radical circle		insimilicenter(circumcircle, excircles radical circle)
circumcircle	first Johnson-Yff circle		-harmonic conjugate of
circumcircle	first Lemoine circle		insimilicenter(circumcircle, 1st Lemoine circle)
circumcircle	Gallatly circle		inverse-in-Brocard-circle of X(1688)
circumcircle	incircle		insimilicenter(circumcircle, incircle)
circumcircle	Lucas inner circle		isogonal conjugate of
circumcircle	Moses circle		harmonic of X(187)
circumcircle	nine-point circle		triangle centroid
circumcircle	orthocentroidal circle		insimilicenter(circumcircle, orthocentroidal circle)
circumcircle	second Brocard circle		circumcenter
circumcircle	second Droz-Farny circle		circumcenter
circumcircle	sine-triple-angle circle		inverse of X(125) in the Brocard circle
circumcircle	Spieker circle		intersection of lines X(1)X(6) and X(2)X(12)
circumcircle	Steiner circle		3/5*og
circumcircle	tangential circle		Exeter point
Conway circle	excircles radical circle		triangle centroid
Conway circle	incircle		incenter
cosine circle	first Lemoine circle		insimilicenter(1st Lemoine circle, 2nd Lemoine circle)
cosine circle	Gallatly circle		inverse-in-Brocard-circle of
cosine circle	incircle		isogonal conjugate of
cosine circle	Lucas circles radical circle		Kenmotu point
cosine circle	Moses circle		insimilicenter(Moses circle, second Lemoine circle)
cosine circle	nine-point circle		Vecten point
excircles radical circle	incircle		insimilicenter(excircles radical circle, incircle)
excircles radical circle	nine-point circle		insimilicenter(excircles radical circle, nine-point circle)
excircles radical circle	Spieker circle		Spieker center
extangents circle	intangents circle		insimilicenter(extangents circle, intangents circle)
first Droz-Farny circle	polar circle		orthocenter
first Droz-Farny circle	second Droz-Farny circle		nine-point center
first Johnson-Yff circle	incircle
first Johnson-Yff circle	second Johnson-Yff circle		incenter
first Johnson-Yff circle	Spieker circle		intersection of and the Euler line
first Lemoine circle	Gallatly circle		Kenmotu point (congruent squares point)
first Lemoine circle	incircle		insimilicenter(incircle, 1st Lemoine circle)
first Lemoine circle	Moses circle		insimilicenter(Moses circle, 1st Lemoine circle)
first Lemoine circle	nine-point circle		insimilicenter(first Lemoine circle, nine-point circle)
first Lemoine circle	orthocentroidal circle		insimilicenter(first Lemoine circle, orthocentroidal circle)
first Lemoine circle	second Brocard circle		insimilicenter(first Lemoine circle, second Brocard circle)
first Lemoine circle	Spieker circle		insimilicenter(first Lemoine circle, Spieker circle)
Gallatly circle	half-Moses circle		Brocard midpoint
Gallatly circle	incircle		insimilicenter(Gallatly circle, incircle)
Gallatly circle	Moses circle		Brocard midpoint
Gallatly circle	nine-point circle		insimilicenter(Gallatly circle, nine-point circle)
Gallatly circle	orthocentroidal circle		insimilicenter(Gallatly circle, orthocentroidal circle)
Gallatly circle	second Brocard circle		insimilicenter(Gallatly circle, second Brocard circle)
Gallatly circle	Spieker circle		insimilicenter(Gallatly circle, Spieker circle)
half-Moses circle	incircle		-isoconjugate of
half-Moses circle	Moses circle		Brocard midpoint
half-Moses circle	Spieker circle		crosspoint of and
incircle	Moses circle		insimilicenter(incircle, Moses circle)
incircle	nine-point circle		-harmonic conjugate of
incircle	orthocentroidal circle		insimilicenter(incircle, orthocentroidal circle)
incircle	second Brocard circle		insimilicenter(incircle, second Brocard circle)
incircle	second Johnson-Yff circle		crosspoint of and
incircle	sine-triple-angle circle		exsimilicenter(incircle, sine-triple-angle circle)
incircle	Spieker circle		triangle centroid
inner Napoleon circle	outer Napoleon circle		triangle centroid
inner Soddy circle	outer Soddy circle		incenter
intangents circle	nine-point circle		perspector of the orthic triangle and intangents triangle
intangents circle	tangential circle
Moses circle	nine-point circle		insimilicenter(Moses circle, nine-point circle)
Moses circle	Spieker circle		insimilicenter(Moses circle, Spieker circle)
nine-point circle	orthocentroidal circle		insimilicenter(nine-point circle, orthocentroidal circle)
nine-point circle	second Brocard circle		insimilicenter(nine-point circle, second Brocard circle)
nine-point circle	second Johnson-Yff circle		insimilicenter(nine-point circle, second Johnson-Yff circle)
nine-point circle	sine-triple-angle circle		Kosnita point
nine-point circle	Spieker circle		complementary conjugate
nine-point circle	Steiner circle		nine-point center
nine-point circle	tangential circle		circumcenter
orthocentroidal circle	second Brocard circle		insimilicenter(orthocentroidal circle, second Brocard circle)
orthocentroidal circle	Spieker circle		insimilicenter(orthocentroidal circle, Spieker circle)
second Brocard circle	second Droz-Farny circle		circumcenter
second Brocard circle	Spieker circle		insimilicenter(second Brocard circle, Spieker circle)
second Johnson-Yff circle	Spieker circle		inverse-in-orthocentroidal-circle of
Spieker circle	tangential mid-arc circle		second mid-arc point of the anticomplementary triangle