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Central Circle

Given a circle expressed in trilinear coordinates by


a central circle is a circle such that 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 (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.

circle Kimberling Kimberling center
Adams' circle *   incenter I
anticomplementary circle third power point orthocenter H
Apollonius circle    
Bevan circle incenter I Bevan point V
Brocard circle triangle centroid G midpoint of the Brocard diameter
circumcircle - 0 circumcenter O
Conway circle   incenter I
cosine circle   symmedian point K
de Longchamps circle third power point de Longchamps point L
Dou circle *   eigencenter of the orthic triangle
Euler-Gergonne-Soddy circle *   *  
excircles radical circle external similitude center of incircle and circumcircle Spieker center
extangents circle *   *  
first Droz-Farny circle *   orthocenter H
first Johnson-Yff circle *   first Johnson-Yff center
first Lemoine circle   midpoint of the Brocard diameter
Fuhrmann circle   Fuhrmann center
Gallatly circle   Brocard midpoint
half-altitude circle *   *  
half-Moses circle *   Brocard midpoint
hexyl circle *   incenter I
incentral circle   *  
incircle   incenter I
inner Napoleon circle first isodynamic point S triangle centroid G
inner Soddy circle *   inner Soddy center
inner Vecten circle *   complement point of the inner Vecten point
intangents circle *   *  
Lester circle *    
Lucas central circle *   *  
Lucas circles radical circle triangle centroid G  
Lucas inner circle incenter I *  
MacBeath circle *   *  
Mandart circle   circumcenter of the extouch triangle
McCay circles radical circle *   *  
mixtilinear circle *   *  
mixtilinear incircles radical circle mittenpunkt midpoint of
Morley's circle *   first Morley center
Moses circle isotomic conjugate of Brocard midpoint
Neuberg circles radical circle *   -Ceva conjugate of
nine-point circle circumcenter O nine-point center
orthocentroidal circle circumcenter O midpoint of GH
outer Napoleon circle second isodynamic point triangle centroid G
outer Soddy circle *   outer Soddy center
outer Vecten circle *   complement point of the outer Vecten point
Parry circle    
polar circle circumcenter O orthocenter H
reflected circle -Ceva conjugate of -Ceva conjugate of
second Brocard circle symmedian point K circumcenter O
second Droz-Farny circle symmedian point K circumcenter O
second Johnson-Yff circle *   second Johnson-Yff center
second Steiner circle *   *  
sine-triple-angle circle *    
Spieker circle *   Spieker center
Stammler circle symmedian point K circumcenter O
Stammler circles radical circle     nine-point center N
Steiner circle *   nine-point center N
Stevanovic circle    
symmedial circle   *  
tangential circle circumcenter O circumcenter of the tangential triangle
tangential mid-arc circle     *  
Taylor circle   Taylor center
third Lemoine circle *   *  
van Lamoen circle *    
Yff central circle *   *  
Yff contact circle *   *  
Yiu circle *   *  

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

Kimberling center circles
incenter I Adams' circle, Conway circle, hexyl circle, incircle
triangle centroid G inner Napoleon circle, outer Napoleon circle
circumcenter O circumcircle, second Brocard circle, second Droz-Farny circle, Stammler circle
orthocenter H anticomplementary circle, polar circle, first Droz-Farny circle
nine-point center N nine-point circle, Stammler circles radical circle, Steiner circle
symmedian point K cosine circle
Spieker center excircles radical circle, Spieker circle
de Longchamps point de Longchamps circle
circumcenter of the tangential triangle tangential circle
Brocard midpoint Gallatly circle, half-Moses circle, Moses circle
Bevan point V Bevan circle
center of the sine-triple-angle circle sine-triple-angle circle
eigencenter of orthic triangle Dou circle
isoperimetric point outer Soddy circle
equal detour point inner Soddy circle
midpoint of the Brocard diameter Brocard circle, first Lemoine circle
-Ceva conjugate of Neuberg circles radical circle
-Ceva conjugate of reflected circle
center of the Parry circle Parry circle
first Morley center Morley's circle

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.




cite this as

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



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