World Library  
Flag as Inappropriate
Email this Article

Nagel point

Article Id: WHEBN0009489914
Reproduction Date:

Title: Nagel point  
Author: World Heritage Encyclopedia
Language: English
Subject: Extouch triangle, Isotomic conjugate, Mandart inellipse, Missing science topics/ExistingMathN, Cevian
Publisher: World Heritage Encyclopedia

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 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.

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 1
  • Trilinear coordinates 2
  • See also 3
  • References 4
  • External links 5

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|,

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

See also


  1. ^
  2. ^
  3. ^

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.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.