Extouch triangle

File:Extouch Triangle and Nagel Point.svg

  Arbitrary triangle △ABC

  Excircles, tangent to the sides of △ABC at Template:Mvar

  Extouch triangle △T_AT_BT_C

  Splitters of the perimeter Template:Mvar; intersect at the Nagel point Template:Mvar

In Euclidean geometry, 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:

$$\begin{array}{rccccc}{T}_A=& 0& :& {\csc }^2\frac{B}{2}& :& {\csc }^2\frac{C}{2}\\ T_B=& {\csc }^2\frac{A}{2}& :& 0& :& {\csc }^2\frac{C}{2}\\ T_C=& {\csc }^2\frac{A}{2}& :& {\csc }^2\frac{B}{2}& :& 0\end{array}$$

or equivalently, where Template:Mvar are the lengths of the sides opposite angles Template:Mvar respectively,

$$\begin{array}{rccccc}{T}_A=& 0& :& \frac{a-b+c}{b}& :& \frac{a+b-c}{c}\\ T_B=& \frac{-a+b+c}{a}& :& 0& :& \frac{a+b-c}{c}\\ T_C=& \frac{-a+b+c}{a}& :& \frac{a-b+c}{b}& :& 0\end{array}$$

Also, with Template:Mvar denoting the semiperimeter of the triangle, the vertices of the extouch triangle are given in barycentric coordinates by:

$$\begin{array}{rccccc}{T}_A=& 0& :& s-b& :& s-c\\ T_B=& s-a& :& 0& :& s-c\\ T_C=& s-a& :& s-b& :& 0\end{array}$$

Related figures

The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram.

The Mandart inellipse is tangent to the sides of the reference triangle at the three vertices of the extouch triangle.

Area

The area of the extouch triangle, Template:Mvar, is given by:

$K_T=K\frac{2r^{2}s}{abc}$

where Template:Mvar and Template:Mvar are the area and radius of the incircle, Template:Mvar is the semiperimeter of the original triangle, and Template:Mvar are the side lengths of the original triangle.

This is the same area as that of the intouch triangle.