Wolfram Researchmathworld.wolfram.comOther Wolfram Sites
Search Site

Applied Mathematics
Calculus and Analysis
Discrete Mathematics
Foundations of Mathematics
History and Terminology
Number Theory
Probability and Statistics
Recreational Mathematics
Alphabetical Index

About MathWorld
About the Author
Terms of Use

What's New
Headline News (RSS)
Random Entry
Live 3D Graphics

Email Comments
Sign the Guestbook

Order book from Amazon

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

Links search


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

mathematica calculationcenter