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

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

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 .

See also Circumcircle Mid-Arc Triangle ,

Fuhrmann Center ,

Fuhrmann
Circle ,

Mid-Arc Points
Explore with Wolfram|Alpha
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

Subject classifications