Nagel point

In geometry, the Nagel point (named for Christian Heinrich von Nagel) 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. It is the point of concurrency of all three of the triangle's splitters.

  Arbitrary triangle △ABC

  Excircles, tangent to the sides of △ABC at T_A, T_B, T_C

  Extouch triangle △T_AT_BT_C

  Splitters of the perimeter AT_A, BT_B, CT_C; intersect at the Nagel point N

Construction

Given a triangle △ABC, let T_A, T_B, T_C be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and the C-excircle meets line AB, respectively. The lines AT_A, BT_B, CT_C concur in the Nagel point N of triangle △ABC.

Another construction of the point T_A is to start at A and trace around triangle △ABC half its perimeter, and similarly for T_B and T_C. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments AT_A, BT_B, CT_C 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 ${\displaystyle s={\tfrac {a+b+c}{2}}}$ is the semi-perimeter of the reference triangle △ABC.

Trilinear coordinates

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

${\displaystyle \csc ^{2}\left({\frac {A}{2}}\right)\,:\,\csc ^{2}\left({\frac {B}{2}}\right)\,:\,\csc ^{2}\left({\frac {C}{2}}\right)}$

or, equivalently, in terms of the side lengths ${\displaystyle a=\left|{\overline {BC}}\right|,b=\left|{\overline {CA}}\right|,c=\left|{\overline {AB}}\right|,}$

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

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]

See also

• Mandart inellipse
• Trisected perimeter point