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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]] | |||
Revision as of 00:39, 4 September 2013
In geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.
Coordinates
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 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.[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 the intouch triangle.
See also
References
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