Nine-Point Center

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


and is the midpoint of the line between the circumcenter O and orthocenter H. The nine-point center is Kimberling center X_5.

It satisfies


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

Distances to some other named triangle centers include


where F is the Feuerbach point, G is the triangle centroid, H is the orthocenter, I is the incenter, L is the de Longchamps point, O is the circumcenter, Sp is the Spieker center, and Delta is the triangle area.

The nine-point center N, vecten point X_(485), and inner vecten point X_(486) 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.

See also

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

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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.", C. "Encyclopedia of Triangle Centers: X(5)=Nine-Point Center.", J. M. "Problema 163."

Referenced on Wolfram|Alpha

Nine-Point Center

Cite this as:

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

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