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# Feuerbach Point

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

 (1)

and is Kimberling center .

If is the Feuerbach point a triangle and , , and are the midpoints of the sides , , and , respectively, then one of the distances , , and is equal to the sum of the two others. For example, in the above figure,

 (2)

Distances to some other named triangle centers include

 (3) (4) (5) (6)

where is the triangle centroid, is the incenter, is the symmedian point, is the circumcenter, is the nine-point center, is the Spieker center, is the triangle area, and is the inradius.

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

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## References

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." http://faculty.evansville.edu/ck6/tcenters/class/feuer.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(11)=Feuerbach Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X11.PandD Software. "Oppervlakte van voetpuntsdriehoeken, voetpuntscirkels." http://www.pandd.demon.nl/voetpdrieh.htm.Pedoe, 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. http://forumgeom.fau.edu/FG2006volume6/FG200621index.html.

Feuerbach Point

## Cite this as:

Weisstein, Eric W. "Feuerbach Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FeuerbachPoint.html