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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.]]
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| In [[geometry]], the '''extouch triangle''' of a [[triangle]] is formed by joining the points at which the three [[excircle]]s touch the triangle.
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| ==Coordinates==
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| The [[vertex (geometry)|vertices]] of the extouch triangle are given in [[trilinear coordinates]] by:
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| :<math>T_A = 0 : \csc^2{\left( B/2 \right)} : \csc^2{\left( C/2 \right)}</math>
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| :<math>T_B = \csc^2{\left( A/2 \right)} : 0 : \csc^2{\left( C/2 \right)}</math>
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| :<math>T_C = \csc^2{\left( A/2 \right)} : \csc^2{\left( B/2 \right)} : 0</math>
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| Or, equivalently, where a,b,c are the lengths of the sides opposite angles A, B, C respectively,
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| :<math>T_A = 0 : \frac{a-b+c}{b} : \frac{a+b-c}{c}</math>
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| :<math>T_B = \frac{-a+b+c}{a} : 0 : \frac{a+b-c}{c}</math>
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| :<math>T_C = \frac{-a+b+c}{a} : \frac{a-b+c}{b} : 0</math>
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| ==Related figures==
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| 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.
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| The [[Mandart inellipse]] is tangent to the sides of the triangles at the three vertices of the extouch triangle.<ref>{{citation
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| | last = Juhász | first = Imre
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| | journal = Annales Mathematicae et Informaticae
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| | mr = 3005114
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| | pages = 37–46
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| | title = Control point based representation of inellipses of triangles
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| | url = http://ami.ektf.hu/uploads/papers/finalpdf/AMI_40_from37to46.pdf
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| | volume = 40
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| | year = 2012}}.</ref>
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| ==Area==
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| The area of the extouch triangle, <math>A_T</math>, is given by:
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| :<math>A_T= A \frac{2r^2s}{abc}</math>
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| 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.
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| This is the same area as the [[intouch triangle]].
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| ==See also==
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| *[[Excircle]]
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| *[[Incircle]]
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| *[[Intouch triangle]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://mathworld.wolfram.com/ExtouchTriangle.html Extouch triangle at MathWorld]
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| [[Category:Circles]]
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| [[Category:Triangle geometry]]
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| [[Category:Triangles]]
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Look at my blog best psychics (My Web Page)