TOPICS

# Fuhrmann Triangle

The Fuhrmann triangle of a reference triangle is the triangle formed by reflecting the mid-arc points , , about the lines , , and .

The Fuhrmann triangle has trilinear vertex matrix

 (1)

The area of the Fuhrmann triangle is given by

 (2) (3)

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

The side lengths are

 (4) (5) (6)

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

Surprisingly, the orthocenter of the Fuhrmann triangle is the incenter of the reference triangle. Furthermore, the nine-point center of the Fuhrmann triangle and are coincident, and the radius of the nine point circle of the Fuhrmann triangle is (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 .

 center of Fuhrmann triangle center of reference triangle circumcenter Fuhrmann center orthocenter incenter nine-point center nine-point center perspector of abc and orthic-of-orthic triangle Zosma transform of Euler infinity point intersection of and Kosnita point anticomplement of Prasolov point circumcenter of extouch triangle Nagel point focus of Kiepert parabola orthocenter Jerabek antipode midpoint of and center of Jerabek hyperbola Spieker center inverse-in-circumcircle of reflection of incenter in Feuerbach point -Ceva conjugate of (,)-harmonic conjugate of reflection of in circumcenter of the orthic triangle Feuerbach point complement of Johnson midpoint isogonal conjugate of isogonal conjugate of isogonal conjugate of isogonal conjugate of isogonal conjugate of crossdifference of and isogonal conjugate of isogonal conjugate of isogonal conjugate of isogonal conjugate of direction of vector , where isogonal conjugate of crossdifference of line and isogonal conjugate of crosspoint of and inverse-in-incircle of isogonal conjugate of isogonal conjugate of orthopoint of direction of vector , where Rigby-Lalescu orthopole (,)-harmonic conjugate of Hatzipolakis reflection point orthocenter of the contact triangle inverse-in-circumcircle of Feuerbach antipode isogonal conjugate of isogonal conjugate of isogonal conjugate of isogonal conjugate of orthic isogonal conjugate of isogonal conjugate of

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

## Explore with Wolfram|Alpha

More things to try:

## 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