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[[Image:Extouch Triangle and Nagel Point.svg|right|frame|325px|The extouch triangle (red, ΔT<sub>A</sub>T<sub>B</sub>T<sub>C</sub>) and the [[Nagel point]] (blue, N) of a triangle (black, ΔABC). 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 [[excircle]]s touch the triangle.

==Coordinates==
The [[vertex (geometry)|vertices]] of the extouch triangle are given in [[trilinear coordinates]] by:

:<math>T_A = 0 : \csc^2{\left( B/2 \right)} : \csc^2{\left( C/2 \right)}</math>
:<math>T_B = \csc^2{\left( A/2 \right)} : 0 : \csc^2{\left( C/2 \right)}</math>
:<math>T_C = \csc^2{\left( A/2 \right)} : \csc^2{\left( B/2 \right)} : 0</math>

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

:<math>T_A = 0 : \frac{a-b+c}{b} : \frac{a+b-c}{c}</math>
:<math>T_B = \frac{-a+b+c}{a} : 0 : \frac{a+b-c}{c}</math>
:<math>T_C = \frac{-a+b+c}{a} :  \frac{a-b+c}{b} : 0</math>

==Related figures==
The intersection of the lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle is the [[Nagel point]]. This is shown in blue and labelled "N" in the diagram.

The [[Mandart inellipse]] is tangent to the sides of the triangles at the three vertices of the extouch triangle.<ref>{{citation
 | last = Juhász | first = Imre
 | journal = Annales Mathematicae et Informaticae
 | mr = 3005114
 | pages = 37–46
 | title = Control point based representation of inellipses of triangles
 | url = http://ami.ektf.hu/uploads/papers/finalpdf/AMI_40_from37to46.pdf
 | volume = 40
 | year = 2012}}.</ref>

==Area==

The area of the extouch triangle, <math>A_T</math>, is given by:

:<math>A_T= A  \frac{2r^2s}{abc}</math>

where <math>A</math>, <math>r</math>, <math>s</math> are the area, radius of the [[incircle]] and [[semiperimeter]] of the original triangle, and <math>a</math>, <math>b</math>, <math>c</math> are the side lengths of the original triangle.

This is the same area as the [[intouch triangle]].

==See also==
*[[Excircle]]
*[[Incircle]]
*[[Intouch triangle]]

==References==
{{reflist}}

==External links==
* [http://mathworld.wolfram.com/ExtouchTriangle.html Extouch triangle at MathWorld]

[[Category:Circles]]
[[Category:Triangle geometry]]
[[Category:Triangles]]