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

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 .

See also Cleavance Center ,

Excenter ,

Excentral Triangle ,

Excircles ,

Fuhrmann Circle ,

Gergonne
Point ,

Mittenpunkt ,

Nagel
Line ,

Splitter ,

Trisected
Perimeter Point
Explore with Wolfram|Alpha
References 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. Referenced on Wolfram|Alpha Nagel Point
Cite this as:
Weisstein, Eric W. "Nagel Point." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/NagelPoint.html

Subject classifications