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Extouch triangle

Triangle formed from the points of tangency of a given triangle's excircles

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated December 3, 2025
Image credit is listed at the end of this article.

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

01Coordinates

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

{\begin{array}{rccccc}T_{A}=&0&:&\csc ^{2}{\frac {B}{2}}&:&\csc ^{2}{\frac {C}{2}}\\T_{B}=&\csc ^{2}{\frac {A}{2}}&:&0&:&\csc ^{2}{\frac {C}{2}}\\T_{C}=&\csc ^{2}{\frac {A}{2}}&:&\csc ^{2}{\frac {B}{2}}&:&0\end{array}}

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

{\begin{array}{rccccc}T_{A}=&0&:&{\frac {a\,-\,b\,+\,c}{b}}&:&{\frac {a\,+\,b\,-\,c}{c}}\\T_{B}=&{\frac {-a\,+\,b\,+\,c}{a}}&:&0&:&{\frac {a\,+\,b\,-\,c}{c}}\\T_{C}=&{\frac {-a\,+\,b\,+\,c}{a}}&:&{\frac {a\,-\,b\,+\,c}{b}}&:&0\end{array}}

Also, with s denoting the semiperimeter of the triangle, the vertices of the extouch triangle are given in barycentric coordinates by:

{\begin{array}{rccccc}T_{A}=&0&:&s-b&:&s-c\\T_{B}=&s-a&:&0&:&s-c\\T_{C}=&s-a&:&s-b&:&0\end{array}}

03Area

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

K_{T}=K{\frac {2r^{2}s}{abc}}

where K and r are the area and radius of the incircle, s is the semiperimeter of the original triangle, and a, b, c are the side lengths of the original triangle.

This is the same area as that of the intouch triangle.

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Sources and credits

This article is adapted from the Wikipedia article “Extouch triangle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

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