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 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 example ...

, one of the points associated with a given

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

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 Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...

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.
Easynagel

Relation to other triangle centers

The Nagel point is the

isotomic conjugate In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sid ...

of the

Gergonne point 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. ...

. 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 ob ...

, 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 bisec ...

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 ...

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 not ...

. 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 In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

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 t ...

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 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 know ...

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 occasiona ...

References

External links

Nagel Point
from

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