Extouch triangle

From Wikipedia, the free encyclopedia

Arbitrary triangle $△ ABC$

  Excircles, tangent to the sides of $△ ABC$ at $T A, T B, T C$

  Extouch triangle $△ T A T B T C$

  Splitters of the perimeter $AT A, BT B, CT C$ ; intersect at the Nagel point $N$

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 $a, b, c$ are the lengths of the sides opposite angles $A, B, C$ 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 $s$ 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

Area

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

K_{T}=K{\frac {2r^{2}s}{abc}}

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

This is the same area as that of the intouch triangle.^[2]

References

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