# Nagel point

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.

## 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;[1][2] equivalently, the Nagel point is the incenter of the anticomplementary triangle.

## Trilinear coordinates

The trilinear coordinates of the Nagel point are[3] as

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

## 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.[4]

## References

- 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.