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Fuhrmann Triangle


FuhrmannTriangle

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].
(1)

The area of the Fuhrmann triangle is given by

Delta_F=(a^3-ba^2-ca^2-b^2a-c^2a+3bca+b^3+c^3-bc^2-b^2c)/((a-b-c)(a+b-c)(a-b+c))Delta
(2)
=-((a+b+c)OI^2)/(4R),
(3)

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

a^'=sqrt(((-a+b+c)(a+b+c))/(bc))OI
(4)
b^'=sqrt(((a-b+c)(a+b+c))/(ca))OI
(5)
c^'=sqrt(((a+b-c)(a+b+c))/(ab))OI.
(6)

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_4orthocenterX_1incenter
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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References

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. https://mathworld.wolfram.com/FuhrmannTriangle.html

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