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Central Circle


Given a circle expressed in trilinear coordinates by

 (lalpha+mbeta+ngamma)(aalpha+bbeta+cgamma)+k(abetagamma+bgammaalpha+calphabeta)=0,

a central circle is a circle such that l:m:n is a triangle center and k is a homogeneous function that is symmetric in the side lengths a, b, and c (Kimberling 1998, p. 226).

The following table summarizes the triangle centers whose trilinears correspond to a circle with l:m:n (for some appropriate value of k). In the table, * indicated a circle function that is known but which does not appear among the list of Kimberling centers. Note also that the circumcircle is not actually a central circle, since its trilinears 0:0:0 are not those of a triangle center.

circleKimberlingl:m:nKimberlingcenter
Adams' circle*X_1incenter I
anticomplementary circleX_(32)third power pointX_4orthocenter H
Apollonius circleX_(940)X_(970)
Bevan circleX_1incenter IX_(40)Bevan point V
Brocard circleX_2triangle centroid GX_(182)midpoint of the Brocard diameter
circumcircle-0X_3circumcenter O
Conway circleX_(213)X_1incenter I
cosine circleX_(69)X_6symmedian point K
de Longchamps circleX_(32)third power pointX_(20)de Longchamps point L
Dou circle*X_(155)eigencenter of the orthic triangle
Euler-Gergonne-Soddy circle**
excircles radical circleX_(56)external similitude center of incircle and circumcircleX_(10)Spieker center
extangents circle**
first Droz-Farny circle*X_4orthocenter H
first Johnson-Yff circle*X_(1478)first Johnson-Yff center
first Lemoine circleX_(141)X_(182)midpoint of the Brocard diameter
Fuhrmann circleX_(48)X_(355)Fuhrmann center
Gallatly circleX_(183)X_(39)Brocard midpoint
GEOS circle**
half-altitude circle**
half-Moses circle*X_(39)Brocard midpoint
hexyl circle*X_1incenter I
incentral circleX_(191)*
incircleX_(220)X_1incenter I
inner Napoleon circleX_(15)first isodynamic point SX_2triangle centroid G
inner Soddy circle*X_(176)inner Soddy center
inner Vecten circle*X_(642)complement of the inner Vecten point
intangents circle**
Johnson triangle circumcircleX_4orthocenter HX_(50)X_(74)-Ceva conjugate of X_(184)
Kenmotu circleX_(371)Kenmotu pointX_(492)Cevapoint of X_2 and X_(488)
Lester circle*X_(1116)
Longuet-Higgins circleX_(962)Longuet-Higgins pointX_(1500)internal similitude center of Moses circle and incircle
Lucas central circle**
Lucas circles radical circleX_2triangle centroid GX_(1151)
Lucas inner circleX_1incenter I*
MacBeath circle**
Mandart circleX_(221)X_(1158)circumcenter of the extouch triangle
McCay circles radical circle**
mixtilinear circle**
mixtilinear incircles radical circleX_9mittenpunktX_(999)midpoint of X_1X_(57)
Morley's circle*X_(356)first Morley center
Moses circleX_(599)isotomic conjugate of X_(598)X_(39)Brocard midpoint
Moses-Longuet-Higgins circle*X_(220)X_9-Ceva conjugate of X_(55)
Neuberg circles radical circle*X_(194)X_6-Ceva conjugate of X_2
nine-point circleX_3circumcenter OX_5nine-point center N
orthocentroidal circleX_3circumcenter OX_(381)midpoint of GH
orthoptic circle of the Steiner inellipseX_3circumcenter OX_2triangle centroid G
outer Napoleon circleX_(16)second isodynamic point S^'X_2triangle centroid G
outer Soddy circle*X_(175)outer Soddy center
outer Vecten circle*X_(641)complement of the outer Vecten point
Parry circleX_(690)X_(351)
polar circleX_3circumcenter OX_4orthocenter H
reflection circleX_(195)X_5-Ceva conjugate of X_3X_(195)X_5-Ceva conjugate of X_3
second Brocard circleX_6symmedian point KX_3circumcenter O
second Droz-Farny circleX_6symmedian point KX_3circumcenter O
second Johnson-Yff circle*X_(1479)second Johnson-Yff center
second Steiner circle**
sine-triple-angle circle*X_(49)
Spieker circle*X_(10)Spieker center Sp
Stammler circleX_6symmedian point KX_3circumcenter O
Stammler circles radical circleX_5nine-point center N
Steiner circle*X_5nine-point center N
Stevanović circleX_(905)X_(650)
symmedial circleX_(2896)*
tangential circleX_3circumcenter OX_(26)circumcenter of the tangential triangle
tangential mid-arc circle*
Taylor circleX_(394)X_(389)Taylor center
third Lemoine circle**
van Lamoen circle*X_(1158)
Yff central circle**
Yff contact circle**
Yiu circle**

The following table summarizes circles sorted by center and indicates concentric circles.


See also

Central Conic, Central Line, Circle, Circle Function

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Central Circle

Cite this as:

Weisstein, Eric W. "Central Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralCircle.html

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