The extouch triangle
is the triangle formed by the points of tangency of a triangle
with its excircles
,
, and
. The points
,
, and
can also be constructed as the points which bisect the perimeter of
starting at
,
, and
.
It is the Cevian triangle of the Nagel point (Kimberling 1998, p. 158), the pedal
triangle of the Bevan point , and the cyclocevian
triangle of
.
It is the polar triangle of the Mandart inellipse.
Equivalent trilinear vertex matrices for the extouch triangle are
(1)
| |||
(2)
|
The side lengths of the extouch triangle are
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
|
where
is the triangle area.
The extouch triangle has area
(9)
| |||
(10)
|
where ,
, and
are the area, inradius, and semiperimeter, respectively, of
the original triangle
.
This is the same area as the contact triangle.
The circumcircle of the extouch triangle is known as the Mandart circle.
The following table gives all centers of the extouch triangle in terms of the centers of the reference triangle that are Kimberling
centers .
center of extouch triangle | center of reference triangle | ||
triangle centroid | |||
circumcenter | circumcenter of extouch triangle |