Nagel point

The Nagel point (blue, N) of a triangle (black). The red triangle is the extouch triangle, and the orange circles are the excircles

In geometry, the Nagel point is a point associated with any 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.

Relation to other triangle centers

The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the incenter, and the centroid are collinear. The incenter is the Nagel point of the medial triangle (Anonymous 1896); equivalently, the Nagel point is the incenter of the anticomplementary triangle.

Trilinear coordinates

Trilinear coordinates of the Nagel point were given by Gallatly (1913) 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|,

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

See also

• Mandart inellipse
• Splitter (geometry)
• Triangle center

References

• Anonymous; Hoover, William; Anthony, O. W (1896). Problem 73. "Geometry: 69-72". American Mathematical Monthly 3 (12): 329. doi:10.2307/2970994. JSTOR 2970994. 
• Baptist, Peter (1987). "Historische Anmerkungen zu Gergonne- und Nagel-Punkt". Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften 71 (2): 230–233. MR0936136. 
• Gallatly, William (1913). The Modern Geometry of the Triangle (2nd ed.). London: Hodgson. pp. page 20. 

External links

• Nagel Point from Cut-the-knot
• Nagel Point, Clark Kimberling
• Weisstein, Eric W., "Nagel Point" from MathWorld.
• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.