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
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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.
See also
Bevan Circle, Excenter, Excenter-Excenter Circle, Excentral-Hexyl Ellipse, Excircles, Extouch Triangle, Gergonne Point, Hexyl Triangle, Mittenpunkt, Soddy CirclesExplore with Wolfram|Alpha
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References
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 TriangleCite this as:
Weisstein, Eric W. "Excentral Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExcentralTriangle.html