Excentral Triangle


The excentral triangle, also called the tritangent triangle, of a triangle DeltaABC is the triangle J=DeltaJ_AJ_BJ_C with vertices corresponding to the excenters of DeltaABC.

It is the anticevian triangle with respect to the incenter I (Kimberling 1998, p. 157), and also the antipedal triangle with respect to I.

The circumcircle of the excentral triangle is the Bevan circle.

Its trilinear vertex matrix is

 [-1  1  1;  1 -1  1;  1  1 -1].

The excentral triangle has side lengths


and area


where Delta, r, and s are the area, inradius, and semiperimeter of the original triangle DeltaABC, respectively. It therefore has the same side lengths and area as the hexyl triangle.

The excentral triangle is perspective to every Cevian triangle (Kimberling 1998, p. 157).

The excentral-hexyl ellipse passes through the vertex of the excentral and hexyl triangles.


Beginning with an arbitrary triangle Delta, find the excentral triangle J. Then find the excentral triangle J^' of that triangle, and so on. Then the resulting triangle J^((infty)) approaches an equilateral triangle (Johnson 1929, p. 185; Goldoni 2003). The analogous result also holds for iterative construction of contact triangles (Goldoni 2003).


Given a triangle DeltaABC, draw the excentral triangle DeltaJ_AJ_BJ_C and medial triangle DeltaM_AM_BM_C. Then the orthocenter H of DeltaABC, incenter I_m of DeltaM_AM_BM_C, and circumcenter O_e of DeltaJ_AJ_BJ_C are collinear with I_m the midpoint of HO_e (Honsberger 1995).


The incenter I of DeltaABC coincides with the orthocenter H_e of DeltaJ_AJ_BJ_C, and the circumcenter O of DeltaABC coincides with the nine-point center N_e of DeltaJ_AJ_BJ_C. Furthermore, N_e=O is the midpoint of the line segment joining the orthocenter H_e and circumcenter O_e of DeltaJ_AJ_BJ_C (Honsberger 1995).

The following table gives the centers of the excentral triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

X_ncenter of excentral triangleX_ncenter of reference triangle
X_1incenterX_(164)incenter of excentral triangle
X_2triangle centroidX_(165)triangle centroid of the excentral triangle
X_3circumcenterX_(40)Bevan point
X_5nine-point centerX_3circumcenter
X_6symmedian pointX_9mittenpunkt
X_7Gergonne pointX_(166)Gergonne point of excentral triangle
X_8Nagel pointX_(167)Nagel point of excentral triangle
X_9mittenpunktX_(168)mittenpunkt of excentral triangle
X_(15)first isodynamic pointX_(1277)third Evans perspector
X_(16)second isodynamic pointX_(1276)second Evans perspector
X_(19)Clawson pointX_(173)congruent isoscelizers point
X_(24)perspector of abc and orthic-of-orthic triangleX_(46)X_4-Ceva conjugate of X_1
X_(25)homothetic center of orthic and tangential trianglesX_(57)isogonal conjugate of X_9
X_(30)Euler infinity pointX_(517)isogonal conjugate of X_(104)
X_(31)second power pointX_(362)congruent circumcircles isoscelizer point
X_(32)third power pointX_(169)X_(85)-Ceva conjugate of X_1
X_(33)perspector of the orthic and intangents trianglesX_(258)congruent incircles isoscelizer point
X_(46)X_4-Ceva conjugate of X_1X_(505)third isoscelizer point
X_(48)crosspoint of X_1 and X_(63)X_(504)second isoscelizer point
X_(51)triangle centroid of orthic triangleX_2triangle centroid
X_(52)orthocenter of orthic triangleX_4orthocenter
X_(53)symmedian point of orthic triangleX_6symmedian point
X_(54)Kosnita pointX_(191)X_(10)-Ceva conjugate of X_1
X_(57)isogonal conjugate of X_9X_(363)equal perimeters isoscelizer point
X_(63)isogonal conjugate of X_(19)X_(845)intersection of lines X_(165)X_(166) and X_(173)X_(503)
X_(64)isogonal conjugate of X_(20)X_(2136)eigentransform of X_(57)
X_(65)orthocenter of the contact triangleX_(188)second mid-arc point of anticomplementary triangle
X_(68)Prasolov pointX_(1490)intersection of lines X_1X_4 and X_3X_9
X_(69)symmedian point of the anticomplementary triangleX_(2951)excentral isogonal conjugate of X_(57)
X_(75)isotomic conjugate of incenterX_(844)intersection of lines X_(166)X_(167) and X_(173)X_(503)
X_(76)third Brocard pointX_(170)X_9-aleph conjugate of X_9
X_(92)Cevapoint of incenter and Clawson pointX_(503)first isoscelizer point
X_(95)Cevapoint of triangle centroid and circumcenterX_(2938)excentral isogonal conjugate of X_2
X_(96)isogonal conjugate of X_(52)X_(2939)excentral isogonal conjugate of X_4
X_(97)isogonal conjugate of X_(53)X_(2941)excentral isogonal conjugate of X_6
X_(98)Tarry pointX_(1282)5th Sharygin point

See also

Bevan Circle, Excenter, Excenter-Excenter Circle, Excentral-Hexyl Ellipse, Excircles, Extouch Triangle, Gergonne Point, Hexyl Triangle, Mittenpunkt, Soddy Circles

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Goldoni, G. "Problem 10993." Amer. Math. Monthly 110, 155, 2003.Honsberger, R. "A Trio of Nested Triangles." §3.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 27-30, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Excentral Triangle

Cite this as:

Weisstein, Eric W. "Excentral Triangle." From MathWorld--A Wolfram Web Resource.

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