Nagel point

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

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.

Construction[]

Given a triangle ABC, let T_A, T_B, and T_C 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 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 ${\displaystyle s={\frac {a+b+c}{2}}}$ 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]

See also[]

• Mandart inellipse
• Trisected perimeter point

References[]

External links[]

• Nagel Point from Cut-the-knot
• Nagel Point, Clark Kimberling
• Weisstein, Eric W. "Nagel Point". MathWorld.
• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.