Wikipedia

Extouch triangle

The extouch triangle (△TATBTC, with red boundary) and the Nagel point (blue, N) of a triangle (△ABC, with black boundary). The orange circles are the excircles of the triangle.

In geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.

The vertices of the extouch triangle are given in trilinear coordinates by:

or equivalently, where a,b,c are the lengths of the sides opposite angles A, B, C respectively,

Related figures

The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram.

The Mandart inellipse is tangent to the sides of the reference triangle at the three vertices of the extouch triangle.[1]

Area

The area of the extouch triangle, , is given by:

where , , are the area, radius of the incircle and semiperimeter of the original triangle, and , , are the side lengths of the original triangle.

This is the same area as that of the intouch triangle.[2]

References

  1. ^ Juhász, Imre (2012), "Control point based representation of inellipses of triangles" (PDF), Annales Mathematicae et Informaticae, 40: 37–46, MR 3005114.
  2. ^ Weisstein, Eric W. "Extouch Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExtouchTriangle.html
This article is copied from an article on Wikipedia® - the free encyclopedia created and edited by its online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of Wikipedia® encyclopedia articles provide accurate and timely information, please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.

Copyright © 2003-2025 Farlex, Inc Disclaimer
All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.