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.

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]

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 $(s-a:s-b:s-c)$ where $s={\tfrac {a+b+c}{2}}$ is the semi-perimeter of the reference triangle $△ ABC$ .

Trilinear coordinates

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

\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.^[5]

References

^ Dussau, Xavier (April 2020). "Elementary construction of the Nagel point". HAL.
^ Anonymous (1896). "Problem 73". Problems for Solution: Geometry. American Mathematical Monthly. 3 (12): 329. doi:10.2307/2970994. JSTOR 2970994.
^ "Why is the Incenter the Nagel Point of the Medial Triangle?". Polymathematics.
^ Gallatly, William (1913). The Modern Geometry of the Triangle (2nd ed.). London: Hodgson. p. 20.
^ Baptist, Peter (1987). "Historische Anmerkungen zu Gergonne- und Nagel-Punkt". Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften. 71 (2): 230–233. MR 0936136.

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.

[1] Dussau, Xavier (April 2020). "Elementary construction of the Nagel point". HAL.

[Anonymous-2] Anonymous (1896). "Problem 73". Problems for Solution: Geometry. American Mathematical Monthly. 3 (12): 329. doi:10.2307/2970994. JSTOR 2970994.

[3] "Why is the Incenter the Nagel Point of the Medial Triangle?". Polymathematics.

[Gallatly-4] Gallatly, William (1913). The Modern Geometry of the Triangle (2nd ed.). London: Hodgson. p. 20.

[5] Baptist, Peter (1987). "Historische Anmerkungen zu Gergonne- und Nagel-Punkt". Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften. 71 (2): 230–233. MR 0936136.

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