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# Nine-Point Center

The nine-point center (sometimes instead denoted ) is the center of the nine-point circle. It has equivalent triangle center functions

 (1) (2) (3)

and is the midpoint of the line between the circumcenter and orthocenter . The nine-point center is Kimberling center .

It satisfies

 (4)

where is the circumradius of the reference triangle and is the circumcenter.

Distances to some other named triangle centers include

 (5) (6) (7) (8) (9) (10) (11)

where is the Feuerbach point, is the triangle centroid, is the orthocenter, is the incenter, is the de Longchamps point, is the circumcenter, is the Spieker center, and is the triangle area.

The nine-point center , vecten point , and inner vecten point are collinear (J. Montes Valderrama, pers. comm., R. Barroso Campos, Apr. 20, 2004).

The nine-point center lies on the Lester circle and is the center of the nine-point circle and Steiner circle. It lies on the Euler line.

The following table summarizes the nine-point centers for named triangles that are Kimberling centers.

 triangle Kimberling nine-point center anticomplementary triangle circumcenter circumcircle mid-arc triangle midpoint of the incenter and circumcenter circumnormal triangle circumcenter circumtangential triangle circumcenter contact triangle inverse-in-incircle of Euler triangle midpoint of and excentral triangle circumcenter first Morley triangle first Morley center first Yff circles triangle (Johnson midpoint) Fuhrmann triangle nine-point center hexyl triangle circumcenter inner Napoleon triangle triangle centroid medial triangle midpoint of and orthic triangle nine-point center of orthic triangle outer Napoleon triangle triangle centroid reference triangle nine-point center second Yff circles triangle (,)-harmonic conjugate of Stammler triangle circumcenter tangential triangle -of-tangential-triangle

Euler Line, Lester Circle, Nine-Point Circle, Nine-Point Conic, Vecten Points

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## References

Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 624, 1970.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, p. 21, 1967.Dixon, R. Mathographics. New York: Dover, pp. 57-58, 1991.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 27-29, 1928.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Nine-Point Center." http://faculty.evansville.edu/ck6/tcenters/class/npcenter.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(5)=Nine-Point Center." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X5.Pedret, J. M. "Problema 163." http://www.aloj.us.es/rbarroso/trianguloscabri/sol/sol163ped.htm.

## Referenced on Wolfram|Alpha

Nine-Point Center

## Cite this as:

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