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Nagel point

## Summary

In geometry, the Nagel point is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. The Nagel point is named after Christian Heinrich von Nagel.

The Nagel point (blue, N) of a triangle (black). The red triangle is the extouch triangle, and the orange circles are the excircles

## Construction

Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively. The lines ATA, BTB, CTC concur in the Nagel point N of triangle ABC.

Another construction of the point TA is to start at A and trace around triangle ABC half its perimeter, and similarly for TB and TC. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments ATA, BTB, CTC 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.[1]

Easy construction of the Nagel point

## Relation to other triangle centers

The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle;[2][3] 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 ${\displaystyle (s-a:s-b:s-c)}$  where ${\displaystyle s={\frac {a+b+c}{2}}}$  is the semi-perimeter of the reference triangle ${\displaystyle ABC}$ .

## Trilinear coordinates

The trilinear coordinates of the Nagel point are[4] as

${\displaystyle \csc ^{2}(A/2)\,:\,\csc ^{2}(B/2)\,:\,\csc ^{2}(C/2)}$

or, equivalently, in terms of the side lengths a = |BC|, b = |CA|, and c = |AB|,

${\displaystyle {\frac {b+c-a}{a}}\,:\,{\frac {c+a-b}{b}}\,:\,{\frac {a+b-c}{c}}.}$

## 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 and Carl Gustav Jacob Jacobi.[5]

1. ^ Dussau, Xavier. "Elementary construction of the Nagel point". HAL.{{cite web}}: CS1 maint: url-status (link)