Nagel point

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The Nagel point of a triangle.

In geometry, the Nagel point is a point associated with any triangle. In the plane of a triangle $\scriptstyle ABC$ with side lengths $\scriptstyle a = |BC|$ , $\scriptstyle b = |CA|$ , and $\scriptstyle c = |AB|$ , let $A'$ , $B'$ , and $C'$ be the points in which the $\scriptstyle A$ -excircle meets line $\scriptstyle BC$ , the $\scriptstyle B$ -excircle meets line $\scriptstyle CA$ , and the $\scriptstyle C$ -excircle meets line $\scriptstyle AB$ , respectively. The lines $A A'$ , $B B'$ , $C C'$ concur in the Nagel point of triangle $A B C$ . 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 $A'$ is to start at $\scriptstyle A$ and trace around triangle $\scriptstyle ABC$ half its perimeter, and similarly for $B'$ and $C'$ . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point.

1 Relation to other triangle centers
2 Trilinear coordinates
3 References
4 External links

[edit] 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.

[edit] 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,

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

[edit] References

Anonymous (1896). "Problem 73". American Mathematical Monthly 3 (12): 329.

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.