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Internal Similitude Center


InternalCenterofSimilitude

In general, the internal similitude center of two circles C_1=C(x_1,r_1) and C_2=C(x_2,r_2) with centers given in Cartesian coordinates is given by

 Si(C_1,C_2)=(r_1x_2+r_2x_1)/(r_1+r_2).
(1)

In trilinear coordinates, the internal center of similitude is given by alpha:beta:gamma, where

alpha=(ar_1(aalpha_1+bbeta_1+cgamma_1)alpha_2+ar_2(aalpha_2+bbeta_2+cgamma_2)alpha_1)/a
(2)
beta=(br_1(aalpha_1+bbeta_1+cgamma_1)beta_2+br_2(aalpha_2+bbeta_2+cgamma_2)beta_1)/b
(3)
gamma=(cr_1(aalpha_1+bbeta_1+cgamma_1)gamma_2+cr_2(aalpha_2+bbeta_2+cgamma_2)gamma_1)/c.
(4)

The incircle and circumcircle of a triangle DeltaABC 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 DeltaABC. It is Kimberling center X_(55) and has equivalent triangle center functions

alpha_(55)=a(b+c-a)
(5)
alpha_(55)=1+cosA
(6)
alpha_(55)=cos^2(1/2A).
(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 DeltaABC 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 DeltaABC.

The following table summarizes the internal similitude centers for a number of named circles.

circle 1circle 2Kimberlinginternal similitude center
Adams circleConway circleX_1incenter
Adams circleincircleX_1incenter
anticomplementary circleBevan circleX_(10)Spieker center
anticomplementary circleBrocard circleX_(2542)insimilicenter(anticomplementary circle, Brocard circle)
anticomplementary circlecircumcircleX_2triangle centroid G
anticomplementary circlecosine circleX_(1587)point Castor I
anticomplementary circlefirst Droz-Farny circleX_4orthocenter H
anticomplementary circlefirst Lemoine circleX_(2546)insimilicenter(anticomplementary circle, first Lemoine circle)
anticomplementary circleGallatly circleX_(2544)insimilicenter(anticomplementary circle, Gallatly circle)
anticomplementary circleincircleX_(388)intersection of lines (X_1,X_4) and (X_7,X_8)
anticomplementary circleMoses circleX_(2548)insimilicenter(anticomplementary circle, Moses circle)
anticomplementary circleorthocentroidal circleX_(2552)insimilicenter(anticomplementary circle, orthocentroidal circle)
anticomplementary circlepolar circleX_4orthocenter H
anticomplementary circlesecond Johnson-Yff circleX_(497)crosspoint of the Nagel point and Gergonne point
anticomplementary circleSpieker circleX_(2550)insimilicenter(anticomplementary circle, Spieker circle)
Apollonius circleBevan circleX_(1695)insimilicenter(Bevan circle, Apollonius circle)
Apollonius circleBrocard circleX_(1693)insimilicenter(Brocard circle, Apollonius circle)
Apollonius circlecircumcircleX_(573)orthoharmonic of X(58)
Apollonius circlecosine circleX_(1685)insimilicenter(2nd Lemoine circle, Apollonius circle)
Apollonius circleexcircles radical circleX_(2538)insimilicenter(Apollonius circle, Brocard circle)
Apollonius circlefirst Lemoine circleX_(1683)insimilicenter(first Lemoine circle, excircles radical circle)
Apollonius circleGallatly circleX_(2019)exsimilicenter(Gallatly circle, Apollonius circle)
Apollonius circleincircleX_(1682)insimilicenter(incircle, Apollonius circle)
Apollonius circlenine-point circleX_(10)Spieker center
Bevan circleBrocard circleX_(1704)insimilicenter(Bevan circle, Brocard circle)
Bevan circlecircumcircleX_(165)centroid of the excentral triangle
Bevan circlecosine circleX_(1702)insimilicenter(Bevan circle, 2nd Lemoine circle)
Bevan circlefirst Lemoine circleX_(1700)insimilicenter(Bevan circle, 1st Lemoine circle)
Bevan circleGallatly circleX_(2017)insimilicenter(Gallatly circle, Bevan circle)
Bevan circleincircleX_(1697)insimilicenter(Bevan circle, incircle)
Bevan circleMoses circleX_(1571)insimilicenter of excentral and Moses circles
Bevan circlenine-point circleX_(1698)insimilicenter(Bevan circle, nine-point circle)
Bevan circlesecond Brocard circleX_(2572)insimilicenter(Bevan circle, second Brocard circle)
Bevan circleSpieker circleX_9mittenpunkt
Brocard circlecircumcircleX_(1340)insimilicenter(circumcircle, Brocard circle)
Brocard circlecosine circleX_(1669)exsimilicenter(Brocard circle, 2nd Lemoine circle)
Brocard circlefirst Lemoine circleX_(182)midpoint of Brocard diameter
Brocard circleGallatly circleX_(2011)insimilicenter(Gallatly circle, Brocard circle)
Brocard circleincircleX_(1674)insimilicenter(incircle, Brocard circle)
Brocard circleMoses circleX_(2033)insimilicenter(Moses circle, Brocard circle)
Brocard circlenine-point circleX_(1348)insimilicenter(nine-point circle, Brocard circle)
Brocard circleorthocentroidal circleX_(2469)insimilicenter(Brocard circle, orthocentroidal circle)
Brocard circlesecond Brocard circleX_(1342)insimilicenter(circumcircle, 1st Lemoine circle)
Brocard circleSpieker circleX_(1678)insimilicenter(Brocard circle, Spieker circle)
circumcirclecosine circleX_(371)Kenmotu point (congruent squares point)
circumcircleexcircles radical circleX_(2536)insimilicenter(circumcircle, excircles radical circle)
circumcirclefirst Johnson-Yff circleX_(12)(X_1,X_5)-harmonic conjugate of X_(11)
circumcirclefirst Lemoine circleX_(1342)insimilicenter(circumcircle, 1st Lemoine circle)
circumcircleGallatly circleX_(1690)inverse-in-Brocard-circle of X(1688)
circumcircleincircleX_(55)insimilicenter(circumcircle, incircle)
circumcircleLucas inner circleX_(1151)isogonal conjugate of X_(1131)
circumcircleMoses circleX_(574)harmonic of X(187)
circumcirclenine-point circleX_2triangle centroid
circumcircleorthocentroidal circleX_(1344)insimilicenter(circumcircle, orthocentroidal circle)
circumcirclesecond Brocard circleX_3circumcenter
circumcirclesecond Droz-Farny circleX_3circumcenter
circumcirclesine-triple-angle circleX_(184)inverse of X(125) in the Brocard circle
circumcircleSpieker circleX_(958)intersection of lines X(1)X(6) and X(2)X(12)
circumcircleSteiner circleX_(631)3/5*og
circumcircletangential circleX_(22)Exeter point
Conway circleexcircles radical circleX_2triangle centroid G
Conway circleincircleX_1incenter
cosine circlefirst Lemoine circleX_(1687)insimilicenter(1st Lemoine circle, 2nd Lemoine circle)
cosine circleGallatly circleX_(1671)inverse-in-Brocard-circle of X_(1343)
cosine circleincircleX_(1124)isogonal conjugate of X_(1123)
cosine circleLucas circles radical circleX_(371)Kenmotu point
cosine circleMoses circleX_(1504)insimilicenter(Moses circle, second Lemoine circle)
cosine circlenine-point circleX_(485)Vecten point
excircles radical circleincircleX_(2534)insimilicenter(excircles radical circle, incircle)
excircles radical circlenine-point circleX_(2540)insimilicenter(excircles radical circle, nine-point circle)
excircles radical circleSpieker circleX_(10)Spieker center
extangents circleintangents circleX_(55)insimilicenter(extangents circle, intangents circle)
first Droz-Farny circlepolar circleX_4orthocenter H
first Droz-Farny circlesecond Droz-Farny circleX_5nine-point center N
first Johnson-Yff circleincircleX_(388)(X_1,X_4) intersection (X_7,X_8)
first Johnson-Yff circlesecond Johnson-Yff circleX_1incenter I
first Johnson-Yff circleSpieker circleX_(377)intersection of (X_7,X_8) and the Euler line
first Lemoine circleGallatly circleX_(371)Kenmotu point (congruent squares point)
first Lemoine circleincircleX_(1672)insimilicenter(incircle, 1st Lemoine circle)
first Lemoine circleMoses circleX_(2035)insimilicenter(Moses circle, 1st Lemoine circle)
first Lemoine circlenine-point circleX_(1676)insimilicenter(first Lemoine circle, nine-point circle)
first Lemoine circleorthocentroidal circleX_(2471)insimilicenter(first Lemoine circle, orthocentroidal circle)
first Lemoine circlesecond Brocard circleX_(2558)insimilicenter(first Lemoine circle, second Brocard circle)
first Lemoine circleSpieker circleX_(1680)insimilicenter(first Lemoine circle, Spieker circle)
Gallatly circlehalf-Moses circleX_(39)Brocard midpoint
Gallatly circleincircleX_(2007)insimilicenter(Gallatly circle, incircle)
Gallatly circleMoses circleX_(39)Brocard midpoint
Gallatly circlenine-point circleX_(2009)insimilicenter(Gallatly circle, nine-point circle)
Gallatly circleorthocentroidal circleX_(2015)insimilicenter(Gallatly circle, orthocentroidal circle)
Gallatly circlesecond Brocard circleX_(2562)insimilicenter(Gallatly circle, second Brocard circle)
Gallatly circleSpieker circleX_(2013)insimilicenter(Gallatly circle, Spieker circle)
half-Moses circleincircleX_(2276)X_2-isoconjugate of X_(985)
half-Moses circleMoses circleX_(39)Brocard midpoint
half-Moses circleSpieker circleX_(1107)crosspoint of X_1 and X_(274)
incircleMoses circleX_(1500)insimilicenter(incircle, Moses circle)
incirclenine-point circleX_(12)(X_1,X_5)-harmonic conjugate of X_(11)
incircleorthocentroidal circleX_(2463)insimilicenter(incircle, orthocentroidal circle)
incirclesecond Brocard circleX_(2564)insimilicenter(incircle, second Brocard circle)
incirclesecond Johnson-Yff circleX_(497)crosspoint of Ge and Na
incirclesine-triple-angle circleX_(2477)exsimilicenter(incircle, sine-triple-angle circle)
incircleSpieker circleX_2triangle centroid G
inner Napoleon circleouter Napoleon circleX_2triangle centroid G
inner Soddy circleouter Soddy circleX_1incenter I
intangents circlenine-point circleX_(33)perspector of the orthic triangle and intangents triangle
intangents circletangential circleX_(55)
Moses circlenine-point circleX_(1506)insimilicenter(Moses circle, nine-point circle)
Moses circleSpieker circleX_(1573)insimilicenter(Moses circle, Spieker circle)
nine-point circleorthocentroidal circleX_(1346)insimilicenter(nine-point circle, orthocentroidal circle)
nine-point circlesecond Brocard circleX_(2566)insimilicenter(nine-point circle, second Brocard circle)
nine-point circlesecond Johnson-Yff circleX_(55)insimilicenter(nine-point circle, second Johnson-Yff circle)
nine-point circlesine-triple-angle circleX_(54)Kosnita point
nine-point circleSpieker circleX_(2886)complementary conjugate X_9
nine-point circleSteiner circleX_5nine-point center N
nine-point circletangential circleX_3circumcenter O
orthocentroidal circlesecond Brocard circleX_(2570)insimilicenter(orthocentroidal circle, second Brocard circle)
orthocentroidal circleSpieker circleX_(2467)insimilicenter(orthocentroidal circle, Spieker circle)
second Brocard circlesecond Droz-Farny circleX_3circumcenter O
second Brocard circleSpieker circleX_(2568)insimilicenter(second Brocard circle, Spieker circle)
second Johnson-Yff circleSpieker circleX_(2478)inverse-in-orthocentroidal-circle of X_(377)
Spieker circletangential mid-arc circleX_(188)second mid-arc point of the anticomplementary triangle

See also

External Similitude Center, Homothetic Center, Midcircle, Similitude Center

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References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(55)=Internal Center of Similitude of Circumcircle and Incircle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X55.

Referenced on Wolfram|Alpha

Internal Similitude Center

Cite this as:

Weisstein, Eric W. "Internal Similitude Center." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InternalSimilitudeCenter.html

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