Nagel point

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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.

Construction

Given a triangle Template:Math, let Template:Mvar be the extouch points in which the Template:Mvar-excircle meets line Template:Mvar, the Template:Mvar-excircle meets line Template:Mvar, and the Template:Mvar-excircle meets line Template:Mvar, respectively. The lines Template:Mvar concur in the Nagel point Template:Mvar of triangle Template:Math.

Another construction of the point Template:Mvar is to start at Template:Mvar and trace around triangle Template:Math half its perimeter, and similarly for Template:Mvar and Template:Mvar. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments Template:Mvar 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.

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; 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 $s=\frac{a+b+c}{2}$ is the semi-perimeter of the reference triangle Template:Math.

Trilinear coordinates

The trilinear coordinates of the Nagel point are as

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

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

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

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.

See also

• Mandart inellipse
• Trisected perimeter point

External links

• Nagel Point from Cut-the-knot
• Nagel Point, Clark Kimberling
• Template:Mathworld
• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.