# 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. 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*. The Nagel point is named after Christian Heinrich von Nagel, a nineteenth-century German mathematician, who wrote about it in 1836.

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*|,

## See also

## References

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## 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.

fr:Cercles inscrit et exinscrits d'un triangle#Point de Nagel