Given a circle expressed in trilinear
coordinates by
a central circle is a circle such that is a triangle center
and
is a homogeneous function that is symmetric in the side lengths , ,
and
(Kimberling 1998, p. 226).
The following table summarizes the triangle centers whose trilinears correspond to a circle with
(for some appropriate value of ). 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 .
The following table summarizes circles sorted by center and indicates concentric
circles.
Kimberling center circles incenter Adams'
circle , Conway circle , hexyl
circle , incircle triangle
centroid inner Napoleon circle , orthoptic
circle of the Steiner inellipse , outer Napoleon
circle circumcenter circumcircle ,
second Brocard circle , second
Droz-Farny circle , Stammler circle orthocenter anticomplementary
circle , Johnson triangle circumcircle ,
polar circle , first
Droz-Farny circle nine-point
center nine-point circle , Stammler
circles radical circle , Steiner circle symmedian
point 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 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 reflection
circle center of the Parry
circle Parry circle first
Morley center Morley's circle midpoint of and orthocentroidal
circle outer
Vecten circle inner
Vecten circle Stevanović
circle Longuet-Higgins
point Longuet-Higgins
circle Apollonius
circle mixtilinear
incircles radical circle Lester
circle Lucas
circles radical circle van
Lamoen circle Mandart
circle first
Johnson-Yff circle second
Johnson-Yff circle
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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