Extouch triangle

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The extouch triangle (red, ΔT_aT_bT_c) and the Nagel point (blue, N) of a triangle (black, ΔABC). The orange circles are the excircles of the triangle.

The extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. The vertices of the extouch triangle are given in trilinear coordinates by:

$T_A = 0 : \csc^2{\left( B/2 \right)} : \csc^2{\left( C/2 \right)}$

$T_B = \csc^2{\left( A/2 \right)} : 0 : \csc^2{\left( C/2 \right)}$

$T_C = \csc^2{\left( A/2 \right)} : \csc^2{\left( B/2 \right)} : 0$

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

$T_A = 0 : \frac{a-b+c}{b} : \frac{a+b-c}{c}$

$T_B = \frac{-a+b+c}{a} : 0 : \frac{a+b-c}{c}$

$T_C = \frac{-a+b+c}{a} : \frac{a-b+c}{b} : 0$

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.

[edit] Area

The area of the extouch triangle, $A T$ , is given by:

$A_T= A \frac{2r^2s}{abc}$

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

This is the same area as the intouch triangle.