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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

[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

			(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

In the nondegenerate case, the Fuhrmann triangle is perspective to the García reflection triangle at the Nagel point (Kimberling).

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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, Germany, 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.Kimberling, C. "Encyclopedia of Triangle Centers: García Reflection Triangle." Preamble to

. https://faculty.evansville.edu/ck6/encyclopedia/ETCPart8.html#X15995.

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Fuhrmann Triangle

Cite this as:

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

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Fuhrmann Triangle

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