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
Kimberling center .
is called the extouch triangle, and its is therefore
the Cevian triangle with respect to the Nagel
, 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
Distances to some other named triangle centers include
is the triangle centroid, is the incenter, is the Gergonne point,
is the nine-point
is the circumcenter, is the Spieker center,
and is the triangle
The Nagel point
Na is also the isotomic
conjugate of the Gergonne point Ge.
complement of the Nagel point is the incenter.
See also Cleavance Center
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References Altshiller-Court, N. New York: Barnes and Noble, pp. 160-164, 1952. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd
ed., rev. enl. 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. New York: Chelsea, p. 53,
Treatise on the Geometry of the Circle and Sphere. Eves, H. W. Boston, MA: Allyn and Bacon, p. 83, 1972. A
Survey of Geometry, rev. ed. Gallatly,
W. "The Nagel Point." §30 in London: Hodgson, p. 20, 1913. The
Modern Geometry of the Triangle, 2nd ed. Honsberger,
R. "The Nagel Point
and the Spieker Circle." §1.4 in Washington, DC: Math.
Assoc. Amer., pp. 5-13, 1995. Episodes
in Nineteenth and Twentieth Century Euclidean Geometry. Johnson, R. A.
Boston, MA: Houghton Mifflin, pp. 184 and 225-226, 1929. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Kimberling,
C. "Central Points and Central Lines in the Plane of a Triangle." Math.
Mag. 67, 163-187, 1994. Kimberling, C. "Nagel Point."
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. Referenced on Wolfram|Alpha Nagel Point
Cite this as:
Weisstein, Eric W. "Nagel Point." From
--A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/NagelPoint.html