Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the extouch triangle of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

is formed by joining the points at which the three

excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

s touch the triangle.

Coordinates

The vertices of the extouch triangle are given in

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

by:

\begin
T_A =& 0 &:& \csc^2 \frac &:& \csc^2 \frac \\
T_B =& \csc^2 \frac &:& 0 &:& \csc^2 \frac \\
T_C =& \csc^2 \frac &:& \csc^2 \frac &:& 0
\end

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

\begin
  T_A =& 0 &:& \frac &:& \frac \\
  T_B =& \frac &:& 0 &:& \frac \\
  T_C =& \frac &:&  \frac &:& 0
\end

Also, with denoting the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...

of the triangle, the vertices of the extouch triangle are given in barycentric coordinates by:

\begin
  T_A =& 0 &:& s-b &:& s-c \\
  T_B =& s-a &:& 0 &:& s-c \\
  T_C =& s-a &:&  s-b &:& 0
\end

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 In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concur ...

. 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, , is given by: :

K_T= K\frac

where and are the area and radius of the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...

, is the

of the original triangle, and are the side lengths of the original triangle. This is the same area as that of the

intouch triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

.Weisstein, Eric W. "Extouch Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExtouchTriangle.html

References

{{reflist Circles Objects defined for a triangle