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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 TA, TB, and TC 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 ATA, BTB, CTC 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. The Nagel point (blue, N) of a triangle (black). The red triangle is the extouch triangle, and the orange circles are the excircles

Another construction of the point TA is to start at A and trace around triangle ABC half its perimeter, and similarly for TB and TC. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments ATA, BTB, CTC 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; equivalently, the Nagel point is the incenter of the anticomplementary triangle.
Trilinear coordinates

The trilinear coordinates of the Nagel point are as

$$\csc^2(A/2)\,:\,\csc^2(B/2)\,:\,\csc^2(C/2)$$

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

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

Mandart inellipse
Trisected perimeter point

References

Anonymous (1896). "Geometry: 69-72". American Mathematical Monthly 3 (12): 329. JSTOR 2970994. |chapter= ignored (help)
"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. pp. page 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.

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.