In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Nagel point (named for
Christian Heinrich von Nagel) is a
triangle center, one of the points associated with a given
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
whose definition does not depend on the placement or scale of the triangle. It is the point of
concurrency of all three of the triangle's
splitters.
Construction
Given a triangle , let be the
extouch points in which the -
excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
meets line , the -excircle meets line , and the -excircle meets line , respectively. The lines
concur in the Nagel point of triangle .
Another construction of the point is to start at and trace around triangle
half its perimeter, and similarly for and . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments are called the triangle's
splitters.
There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at the Nagel point.
Relation to other triangle centers
The Nagel point is the
isotomic conjugate of the
Gergonne point. The Nagel point, the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
, and the
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
are
collinear on a line called the ''Nagel line''. The incenter is the Nagel point of the
medial triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...
;
equivalently, the Nagel point is the incenter of the
anticomplementary triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is ...
. The
isogonal conjugate of the Nagel point is the point of concurrency of the lines joining the mixtilinear touchpoint and the opposite vertex.
Barycentric coordinates
The un-normalized
barycentric coordinates of the Nagel point are
where
is the semi-perimeter of the reference triangle .
Trilinear coordinates
The
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
of the Nagel point are
[
] as
:
or, equivalently, in terms of the side lengths
:
History
The Nagel point is named after
Christian Heinrich von Nagel, a nineteenth-century German mathematician, who wrote about it in 1836.
Early contributions to the study of this point were also made by
August Leopold Crelle
August Leopold Crelle (17 March 1780 – 6 October 1855) was a German mathematician. He was born in Eichwerder near Wriezen, Brandenburg, and died in Berlin. He is the founder of ''Journal für die reine und angewandte Mathematik'' (also kn ...
and
Carl Gustav Jacob Jacobi.
See also
*
Mandart inellipse
In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitter ...
*
Trisected perimeter point
In geometry, given a triangle ''ABC'', there exist unique points ''A´'', ''B´'', and ''C´'' on the sides ''BC'', ''CA'', ''AB'' respectively, such that:
:* ''A´'', ''B´'', and ''C´'' partition the perimeter of the triangle into three equa ...
References
External links
Nagel Pointfrom
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Nagel Point Clark Kimberling
* {{mathworld , title = Nagel Point , urlname = NagelPoint
a
Generalizes Spieker circle and associated Nagel line.
Triangle centers
fr:Cercles inscrit et exinscrits d'un triangle#Point de Nagel