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External Similitude Center


ExternalCenterofSimilitude

In general, the external 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

 Se(C_1,C_2)=(r_1x_2-r_2x_1)/(r_2-r_1).
(1)

In trilinear coordinates, the external 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 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 DeltaABC. It is Kimberling center X_(56) and has equivalent triangle center functions

alpha_(56)=a/(b+c-a)
(5)
alpha_(56)=1-cosA
(6)
alpha_(56)=sin^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, the internal similitude center Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle DeltaABC 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 DeltaABC.

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

circle 1circle 2Kimberlingexternal similitude center
Adams' circleConway circleX_1incenter
Adams' circleincircleX_1incenter
anticomplementary circleBevan circleX_(516)isogonal conjugate of X_(103)
anticomplementary circleBrocard circleX_(2543)external similitude center( anticomplementary circle, Brocard circle)
anticomplementary circlecircumcircleX_(20)de Longchamps point L
anticomplementary circlecosine circleX_(1588)point Castor II
anticomplementary circlefirst Droz-Farny circleX_4orthocenter H
anticomplementary circlefirst Lemoine circleX_(2547)external similitude center(anticomplementary circle, first Lemoine circle)
anticomplementary circlefirst Johnson-Yff circleX_(388)intersection of (X_1,X_4) and (X_7,X_8)
anticomplementary circleGallatly circleX_(2545)external similitude center(anticomplementary circle, Gallatly circle)
anticomplementary circleincircleX_(497)crosspoint of the Gergonne point Ge and Nagel point Na
anticomplementary circleMoses circleX_(2549)external similitude center(anticomplementary circle, Moses circle)
anticomplementary circlenine-point circleX_2triangle centroid G
anticomplementary circleorthocentroidal circleX_(2553)external similitude center(anticomplementary circle, orthocentroidal circle)
anticomplementary circlepolar circleX_4orthocenter H
anticomplementary circleSpieker circleX_(2551)external similitude center(anticomplementary circle, Spieker center)
anticomplementary circleSteiner circleX_(20)de Longchamps point L
anticomplementary circletangential triangleX_(23)far-out point
Apollonius circleBevan circleX_(43)X(6)-Ceva conjugate of X(1)
Apollonius circleBrocard circleX_(1694)external similitude center(Brocard circle, Apollonius circle)
Apollonius circlecircumcircleX_(386)inverse-in-Brocard-circle of X(58)
Apollonius circlecosine circleX_(1686)external similitude center(2nd Lemoine circle, Apollonius circle)
Apollonius circleexcircles radical circleX_(2539)external similitude center(Apollonius circle, excircles radical circle)
Apollonius circlefirst Lemoine circleX_(1684)external similitude center(1st Lemoine circle, Apollonius circle)
Apollonius circleGallatly circleX_(2020)insimilicenter(Gallatly circle, Apollonius circle)
Apollonius circleincircleX_(181)Apollonius point
Apollonius circlenine-point circleX_(2051)external similitude center(nine-point circle, Apollonius circle)
Bevan circleBrocard circleX_(1705)external similitude center(Bevan circle, Brocard circle)
Bevan circlecircumcircleX_1incenter
Bevan circlecosine circleX_(1703)external similitude center(Bevan circle, 2nd Lemoine circle)
Bevan circlefirst Lemoine circleX_(1701)external similitude center(Bevan circle, 1st Lemoine circle)
Bevan circleGallatly circleX_(2018)external similitude center(Gallatly circle, Bevan circle)
Bevan circleincircleX_(57)isogonal conjugate of X(9)
Bevan circleMoses circleX_(1572)external similitude center of excentral and Moses circles
Bevan circlenine-point circleX_(1699)external similitude center(Bevan circle, nine-point circle)
Bevan circleSpieker circleX_(1706)external similitude center(Bevan circle, Spieker circle)
Brocard circlecircumcircleX_(1341)external similitude center(circumcircle, Brocard circle)
Brocard circlecosine circleX_(1668)insimilicenter(Brocard circle, 2nd Lemoine circle)
Brocard circlefirst Lemoine circleX_(182)midpoint of Brocard diameter
Brocard circleGallatly circleX_(2012)external similitude center(Gallatly circle, Brocard circle)
Brocard circleincircleX_(1675)external similitude center(incircle, Brocard circle)
Brocard circleMoses circleX_(2034)external similitude center(Moses circle, Brocard circle)
Brocard circlenine-point circleX_(1349)external similitude center(nine-point circle, Brocard circle)
Brocard circleorthocentroidal circleX_(2470)external similitude center(Brocard circle, orthocentroidal circle)
Brocard circlesecond Brocard circleX_(1343)external similitude center(circumcircle, 1st Lemoine circle)
Brocard circleSpieker circleX_(1679)external similitude center(Brocard circle, Spieker circle)
circumcirclecosine circleX_(372)(X(3), X(6))-harmonic conjugate of X(371)
circumcirclefirst Lemoine circleX_(1343)external similitude center(circumcircle, 1st Lemoine circle)
circumcircleGallatly circleX_(1689)inverse-in-Brocard-circle of X(1687)
circumcirclehalf-Moses circleX_6symmedian point K
circumcircleincircleX_(56)external similitude center(circumcircle, incircle)
circumcircleMoses circleX_(32)3rd power point
circumcirclenine-point circleX_4orthocenter
circumcircleorthocentroidal circleX_(1345)external similitude center(circumcircle, orthocentroidal circle)
circumcirclesecond Brocard circleX_3circumcenter
circumcirclesecond Droz-Farny circleX_3circumcenter
circumcirclesecond Johnson-Yff circleX_(11)Feuerbach point F
circumcirclesine-triple-angle circleX_(1147)isogonal conjugate of X(847)
circumcircleSteiner circleX_(20)de Longchamps point
circumcircletangential circleX_(22)Exeter point
Conway circleexcircles radical circleX_8Nagel point Na
Conway circleincircleX_1incenter
cosine circlefirst Lemoine circleX_(1688)external similitude center(1st Lemoine circle, 2nd Lemoine circle)
cosine circleGallatly circleX_(1670)inverse-in-Brocard-circle of X(1342)
cosine circleincircleX_(1335)(X(1), X(6))-harmonic conjugate of X(1124)
cosine circleLucas circles radical circleX_3circumcenter O
cosine circleMoses circleX_(1505)external similitude center(Moses circle, 2nd Lemoine circles)
cosine circlenine-point circleX_(486)inner Vecten point
cosine circleorthocentroidal circleX_(2466)external similitude center(cosine circle, orthocentroidal circle)
cosine circlesecond Brocard circleX_(2561)external similitude center(cosine circle, second Brocard circle)
cosine circleSpieker circleX_(1378)external similitude center(cosine circle, Spieker circle)
excircles radical circleincircleX_(2535)external similitude center(excircles radical circle, incircle)
excircles radical circlenine-point circleX_(2541)external similitude center(excircles radical circle, nine-point circle)
excircles radical circleSpieker circleX_(10)Spieker center
extangents circlenine-point circleX_(19)Clawson point
extangents circletangential circleX_(55)internal similitude center(circumcircle, incircle)
first Droz-Farny circlepolar circleX_4orthocenter H
first Droz-Farny circlesecond Droz-Farny circleX_(30)Euler infinity point
first Lemoine circleGallatly circleX_(372)(X(3), X(6))-harmonic conjugate of X(371)
first Lemoine circleincircleX_(1673)external similitude center(incircle, 1st Lemoine circle)
first Lemoine circleMoses circleX_(2036)external similitude center(Moses circle, 1st Lemoine circle)
first Lemoine circlenine-point circleX_(1677)external similitude center(first Lemoine circle, nine-point circle)
first Lemoine circleorthocentroidal circleX_(2472)external similitude center(first Lemoine circle, orthocentroidal circle)
first Lemoine circlesecond Brocard circleX_(2559)external similitude center(first Lemoine circle, second Brocard circle)
first Lemoine circleSpieker circleX_(1681)external similitude center(first Lemoine circle, Spieker circle)
first Johnson-Yff circleincircleX_4orthocenter H
first Johnson-Yff circlenine-point circleX_(56)external similitude center(circumcircle, incircle)
first Johnson-Yff circlesecond Johnson-Yff circleX_4orthocenter H
Gallatly circlehalf-Moses circleX_(39)Brocard midpoint Omega^('')
Gallatly circleincircleX_(2008)external similitude center(Gallatly circle, incircle)
Gallatly circleMoses circleX_(39)Brocard midpoint
Gallatly circlenine-point circleX_(2010)external similitude center(Gallatly circle, nine-point circle)
Gallatly circleorthocentroidal circleX_(2016)external similitude center(Gallatly circle, orthocentroidal circle)
Gallatly circlesecond Brocard circleX_(2563)external similitude center(Gallatly circle, second Brocard circle)
Gallatly circleSpieker circleX_(2014)external similitude center(Gallatly circle, Spieker circle)
half-Moses circleincircleX_(2275)X_2-isoconjugate of X_(983)
half-Moses circleMoses circleX_(39)Brocard midpoint
half-Moses circleSpieker circleX_(1575)external similitude center(half-Moses circle, Spieker circle)
incircleinner Soddy circleX_(482)second Eppstein point
incircleMoses circleX_(1015)external similitude center(Moses circle, incircle)
incirclenine-point circleX_(11)Feuerbach point
incircleorthocentroidal circleX_(2464)external similitude center(incircle, orthocentroidal circle)
incircleouter Soddy circleX_(481)first Eppstein point
incirclesecond Brocard circleX_(2565)external similitude center(incircle, second Brocard circle)
incirclesecond Johnson-Yff circleX_4orthocenter H
incirclesine-triple-angle circleX_(215)X(1)-Ceva conjugate of X(50)
incircleSpieker circleX_8Nagel point
incircletangential mid-arc circleX_(2089)third mid-arc point
inner Napoleon circleouter Napoleon circleX_2triangle centroid G
inner Soddy circleOuter Soddy circleX_7Gergonne point Ge
intangents circlenine-point circleX_(1062)intersection of (X_1,X_3) and (X_5,X_33)
Moses circlenine-point circleX_(115)center of the Kiepert hyperbola
Moses circleSpieker circleX_(1574)external similitude center(Moses circle, Spieker circle)
nine-point circleorthocentroidal circleX_(1347)external similitude center(nine-point circle, orthocentroidal circle)
nine-point circlesecond Brocard circleX_(2567)external similitude center(nine-point circle, second Brocard circle)
nine-point circlesine-triple-angle circleX_(110)focus of the Kiepert parabola
nine-point circleSpieker circleX_(1329)complementary conjugate of X(1)
nine-point circleSteiner circleX_5nine-point center
nine-point circletangential circleX_3circumcenter
orthocentroidal circlesecond Brocard circleX_(2571)external similitude center(orthocentroidal circle, second Brocard circle)
orthocentroidal circleSpieker circleX_(2468)external similitude center(orthocentroidal circle, second Brocard circle)
second Brocard circlesecond Droz-Farny circleX_3circumcenter
second Brocard circleSpieker circleX_(2569)external similitude center(second Brocard circle, Spieker circle)
sine-triple-angle circletangential circleX_(1993)orthocorrespondent of X_3

See also

Circle-Circle Tangents, Homothetic Center, Internal Similitude 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(56)=External Center of Similitude of Circumcircle and Incircle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X56.

Referenced on Wolfram|Alpha

External Similitude Center

Cite this as:

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

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