Feuerbach Point


The point F at which the incircle and nine-point circle are tangent. It has triangle center function


and is Kimberling center X_(11).


If F is the Feuerbach point a triangle DeltaABC and X, Y, and Z are the midpoints of the sides BC, CA, and AB, respectively, then one of the distances |FX|, |FY|, and |FZ| is equal to the sum of the two others. For example, in the above figure,


Distances to some other named triangle centers include


where G is the triangle centroid, I is the incenter, K is the symmedian point, O is the circumcenter, N is the nine-point center, Sp is the Spieker center, Delta is the triangle area, and r is the inradius.

See also

Feuerbach Antipode, Feuerbach's Theorem, Feuerbach Triangle, Incircle, Nine-Point Circle

Explore with Wolfram|Alpha


Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 200, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Feuerbach Point.", C. "Encyclopedia of Triangle Centers: X(11)=Feuerbach Point." Software. "Oppervlakte van voetpuntsdriehoeken, voetpuntscirkels.", D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.Salmon, G. Conic Sections, 6th ed. New York: Chelsea, p. 127, 1960.Suceava, B. and Yiu, P. "The Feuerbach Point and Euler Lines." Forum Geom. 6, 191-197, 2006.

Referenced on Wolfram|Alpha

Feuerbach Point

Cite this as:

Weisstein, Eric W. "Feuerbach Point." From MathWorld--A Wolfram Web Resource.

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