Fuhrmann Triangle


The Fuhrmann triangle of a reference triangle DeltaABC is the triangle DeltaF_CF_BF_A formed by reflecting the mid-arc points arcM_A, arcM_B, arcM_C about the lines AB, AC, and BC.

The Fuhrmann triangle has trilinear vertex matrix

 [a (-a^2+c^2+bc)/b (-a^2+b^2+bc)/c; (-b^2+c^2+ac)/a b a^2-b^2+ac; (b^2-c^2+ab)/a (a^2-c^2+ab)/b c].

The area of the Fuhrmann triangle is given by


where Delta is the area of the reference triangle, OI is the distance between the circumcenter and incenter of the reference triangle, and R is the circumradius of the reference triangle (P. Moses, pers. comm., Aug. 18, 2005).

The side lengths are


The circumcircle of the Fuhrmann triangle is called the Fuhrmann circle, and the lines F_AM_A, F_BM_B, and F_CM_C concur at the circumcenter O.

Surprisingly, the orthocenter of the Fuhrmann triangle is the incenter of the reference triangle. Furthermore, the nine-point center of the Fuhrmann triangle and DeltaABC are coincident, and the radius of the nine point circle of the Fuhrmann triangle is OI/2 (P. Moses, pers. comm., Aug. 18, 2005).

The following table gives the centers of the Fuhrmann triangle in terms of the centers of the reference triangle that correspond to Kimberling centers X_n.

X_ncenter of Fuhrmann triangleX_ncenter of reference triangle
X_3circumcenterX_(355)Fuhrmann center
X_5nine-point centerX_5nine-point center
X_(24)perspector of abc and orthic-of-orthic triangleX_(1837)Zosma transform of X_(34)
X_(30)Euler infinity pointX_(952)intersection of X_1X_5 and X_3X_8
X_(54)Kosnita pointX_(2475)anticomplement of X_(21)
X_(68)Prasolov pointX_(1158)circumcenter of extouch triangle
X_(74)X_(74)X_8Nagel point
X_(110)focus of Kiepert parabolaX_4orthocenter
X_(113)Jerabek antipodeX_(946)midpoint of X_1 and X_4
X_(125)center of Jerabek hyperbolaX_(10)Spieker center
X_(186)inverse-in-circumcircle of X_4X_(80)reflection of incenter in Feuerbach point
X_(235)X_4-Ceva conjugate of X_(185)X_(496)(X_1,X_5)-harmonic conjugate of X_(495)
X_(265)reflection of X_3 in X_(125)X_3circumcenter
X_(403)X_(36) of the orthic triangleX_(11)Feuerbach point
X_(427)complement of X_(22)X_(495)Johnson midpoint
X_(511)isogonal conjugate of X_(98)X_(2801)isogonal conjugate of X_(2717)
X_(520)isogonal conjugate of X_(107)X_(2827)isogonal conjugate of X_(2743)
X_(523)isogonal conjugate of X_(110)X_(900)crossdifference of X_6 and X_(101)
X_(525)isogonal conjugate of X_(112)X_(2826)isogonal conjugate of X_(2742)
X_(526)isogonal conjugate of X_(476)X_(513)isogonal conjugate of X_(100)
X_(542)direction of vector ax+bx+cx, where X=X_(98)X_(516)isogonal conjugate of X_(103)
X_(690)crossdifference of line X_6 and X_(110)X_(514)isogonal conjugate of X_(101)
X_(1112)crosspoint of X_4 and X_(250)X_(942)inverse-in-incircle of X_(36)
X_(1154)isogonal conjugate of X_(1141)X_(2771)isogonal conjugate of X_(2687)
X_(1503)orthopoint of X_(525)X_(528)direction of vector ax+bx+cx, where X=X_(11)
X_(1594)Rigby-Lalescu orthopoleX_(12)(X_1,X_5)-harmonic conjugate of X_(11)
X_(1986)Hatzipolakis reflection pointX_(65)orthocenter of the contact triangle
X_(2072)inverse-in-circumcircle of X_(26)X_(119)Feuerbach antipode
X_(2777)isogonal conjugate of X_(2693)X_(519)isogonal conjugate of X_(106)
X_(2781)isogonal conjugate of X_(2697)X_(518)isogonal conjugate of X_(105)
X_(2914)orthic isogonal conjugate of X_(186)X_(79)isogonal conjugate of X_(35)

See also

Circumcircle Mid-Arc Triangle, Fuhrmann Center, Fuhrmann Circle, Mid-Arc Points

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Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952.Fuhrmann, W. Synthetische Beweise Planimetrischer Sätze. Berlin, p. 107, 1890.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 228-229, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Fuhrmann Triangle

Cite this as:

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

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