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 Christian Heinrich von Nagel (28 February 1803 in Stuttgart, Germany – 27 October 1882 in Ulm, Germany) was a German geometer. After attending the gymnasium, Nagel went in 1817 to Evangelical Seminaries of Maulbronn and Blaubeuren. From 182 ...

) is a

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For examp ...

, 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- colli ...

whose definition does not depend on the placement or scale of the triangle. It is the point of

concurrency Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

of all three of the triangle's

splitters Splitter or splitters may refer to: Technology * DSL filter or DSL splitter, in telecommunications * Fiber-optic splitter * Hybrid coil, a three windings transformer * Power dividers and directional couplers, in RF engineering * Siamese connect ...

Construction

Given a triangle , let be the extouch points in which the - excircle 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

. 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.
Easynagel

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 bise ...

are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

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 no ...

; equivalently, the Nagel point is the incenter of the anticomplementary triangle. 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

(s-a:s-b:s-c)

where

s = \tfrac

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 ...

of the Nagel point are as :

\csc^2\left(\frac\right)\,:\,\csc^2\left(\frac\right)\,:\,\csc^2\left(\frac\right)

or, equivalently, in terms of the side lengths

a=\left, \overline\, b=\left, \overline\, c=\left, \overline\,

\frac\,:\,\frac\,:\,\frac.

History

The Nagel point is named after

, 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 and

Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasio ...

References

External links

Nagel Point
from Cut-the-knot
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