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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

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

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