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
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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.
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Cite this as:
Weisstein, Eric W. "Nagel Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NagelPoint.html