Let be the point at which the -excircle meets the side of a triangle , and define and similarly. Then the lines , , and concur in the Nagel point (sometimes denoted ). The Nagel point has triangle center function

(1)

and is Kimberling center .

The triangle is called the extouch triangle, and its is therefore the Cevian triangle with respect to the Nagel point.

The points , , and can also be constructed as the points which bisect the perimeter of starting at , , and . For this reason, the Nagel point is sometimes known as the bisected perimeter point (Bennett et al. 1988, Chen et al. 1992, Kimberling 1994), although the cleavance center is also a bisected perimeter point.

The Nagel point lies on the Nagel line. The orthocenter and Nagel point form a diameter of the Fuhrmann circle.

Distances to some other named triangle centers include

			(2)
NaGe			(3)
			(4)
			(5)
			(6)
NaSp			(7)

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

The Nagel point Na is also the isotomic conjugate of the Gergonne point Ge.

The complement of the Nagel point is the incenter.

Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 160-164, 1952.

Bennett, G.; Glenn, J.; Kimberling, C.; and Cohen, J. M. "Problem E 3155 and Solution." Amer. Math. Monthly 95, 874, 1988.

Chen, J.; Lo, C.-H.; and Lossers, O. P. "Problem E 3397 and Solution." Amer. Math. Monthly 99, 70-71, 1992.

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 53, 1971.

Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, p. 83, 1972.

Gallatly, W. "The Nagel Point." §30 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 20, 1913.

Honsberger, R. "The Nagel Point and the Spieker Circle." §1.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 5-13, 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 184 and 225-226, 1929.

Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Kimberling, C. "Nagel Point." http://faculty.evansville.edu/ck6/tcenters/class/nagel.html.

Kimberling, C. "Encyclopedia of Triangle Centers: X(8)=Nagel Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X8.

Nagel, C. H. Untersuchungen über die wichtigsten zum Dreiecke gehöhrigen Kreise. Eine Abhandlung aus dem Gebiete der reinen Geometrie. Leipzig, Germany, 1836.

CITE THIS AS:

Weisstein, Eric W. "Nagel Point." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NagelPoint.html