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# Excentral Triangle

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

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

The circumcircle of the excentral triangle is the Bevan circle.

 (1)

The excentral triangle has side lengths

 (2) (3) (4)

and area

 (5) (6)

where , , and are the area, inradius, and semiperimeter of the original triangle , 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 , find the excentral triangle . Then find the excentral triangle of that triangle, and so on. Then the resulting triangle 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 , draw the excentral triangle and medial triangle . Then the orthocenter of , incenter of , and circumcenter of are collinear with the midpoint of (Honsberger 1995).

The incenter of coincides with the orthocenter of , and the circumcenter of coincides with the nine-point center of . Furthermore, is the midpoint of the line segment joining the orthocenter and circumcenter of (Honsberger 1995).

The following table gives the centers of the excentral triangle in terms of the centers of the reference triangle for Kimberling centers with .

 center of excentral triangle center of reference triangle incenter incenter of excentral triangle triangle centroid triangle centroid of the excentral triangle circumcenter Bevan point orthocenter incenter nine-point center circumcenter symmedian point mittenpunkt Gergonne point Gergonne point of excentral triangle Nagel point Nagel point of excentral triangle mittenpunkt mittenpunkt of excentral triangle first isodynamic point third Evans perspector second isodynamic point second Evans perspector Clawson point congruent isoscelizers point perspector of abc and orthic-of-orthic triangle -Ceva conjugate of homothetic center of orthic and tangential triangles isogonal conjugate of Euler infinity point isogonal conjugate of second power point congruent circumcircles isoscelizer point third power point -Ceva conjugate of perspector of the orthic and intangents triangles congruent incircles isoscelizer point -Ceva conjugate of third isoscelizer point crosspoint of and second isoscelizer point triangle centroid of orthic triangle triangle centroid orthocenter of orthic triangle orthocenter symmedian point of orthic triangle symmedian point Kosnita point -Ceva conjugate of isogonal conjugate of equal perimeters isoscelizer point isogonal conjugate of intersection of lines and isogonal conjugate of eigentransform of orthocenter of the contact triangle second mid-arc point of anticomplementary triangle Prasolov point intersection of lines and symmedian point of the anticomplementary triangle excentral isogonal conjugate of isotomic conjugate of incenter intersection of lines and third Brocard point -aleph conjugate of Cevapoint of incenter and Clawson point first isoscelizer point Cevapoint of triangle centroid and circumcenter excentral isogonal conjugate of isogonal conjugate of excentral isogonal conjugate of isogonal conjugate of excentral isogonal conjugate of Tarry point 5th Sharygin point

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

## 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 Triangle

## Cite this as:

Weisstein, Eric W. "Excentral Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExcentralTriangle.html