TOPICS
Search

Fuhrmann Circle


FuhrmannCircle

The circumcircle of the Fuhrmann triangle. It has the line HNa, where H is the orthocenter and Na is the Nagel point, as its diameter. In fact, these points (Kimberling centers X_4 and X_8, respectively), are the only Kimberling centers lying on it.

While H and Na are the only noteworthy triangle centers to lie on the circle, a total of at least six other noteworthy points lie on the Fuhrmann circle (Honsberger 1995, p. 49). Three of these are the points T, U, and V which are a distance 2r along the altitudes from the vertices, where r is the inradius of DeltaABC (Honsberger 1995, p. 52).

The Fuhrmann circle has circle function

 l=-(2acosA)/(a+b+c),
(1)

corresponding to Kimberling center X_(48) for an appropriate choice of the circle constant k. Its center is known as the Fuhrmann center Fu, and is Kimberling center X_(355). Its radius is

R_F=OI
(2)
=sqrt((a^3-a^2b-ab^2+b^3-a^2c+3abc-b^2c-ac^2+c^3)/(abc))R,
(3)

where OI is the distance between the circumcenter and incenter (P. Moses, pers. comm., May 9, 2005) and R is the circumradius of the reference triangle.

FuhrmannCircleParallel

Interestingly, the segment OI is parallel to the diameter HNa of the Fuhrmann circle (P. Moses, pers. comm., May 9, 2005). Even more amazingly, the two parallelograms OIFuNa and OIHFu have as their centroids the Spieker center and nine-point center, respectively, where IFu=2NI or IFu=R-2r (F. M. Jackson, pers. comm., Apr. 26, 2007).


See also

Altitude, Fuhrmann Center, Fuhrmann Triangle, Inradius, Mid-Arc Points, Nagel Point, Orthocenter

Explore with Wolfram|Alpha

References

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 58, 1971.Fuhrmann, W. Synthetische Beweise Planimetrischer Sätze. Berlin, p. 107, 1890.Honsberger, R. "The Fuhrmann Circle." Ch. 6 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 49-52, 1995.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 Circle

Cite this as:

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

Subject classifications