Extouch triangle: Difference between revisions

Revision as of 21:24, 23 August 2007

Revision as of 21:52, 23 August 2007

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The '''extouch triangle''' of a triangle is formed by joining the points at which the three [[excircle]]s touch the triangle. The 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>

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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==Area==

The area of the extouch triangle, A<submath>TA_T</submath>, 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]].