The contact triangle of a triangle 
, also called the intouch triangle, is the triangle 
formed by the points of
tangency of the incircle of 
with 
.
The contact triangle is therefore the pedal triangle of 
with respect to the incenter 
of 
. It is also the Cevian
triangle of 
with respect to the Gergonne point Ge (Kimberling
1998, p. 158) and the cyclocevian triangle
with respect to the same point.
The contact triangle is the polar triangle of the incircle.
The contact triangle has equivalent trilinear vertex matrices
 |  | ![[0 (ac)/(a-b+c) (ab)/(a+b-c); (bc)/(-a+b+c) 0 (ab)/(a+b-c); (bc)/(-a+b+c) (ac)/(a-b+c) 0]](/images/equations/ContactTriangle/Inline11.svg) |
(1)
|
 |  | ![[0 sec^2(1/2B) sec^2(1/2C); sec^2(1/2A) 0 sec^2(1/2C); sec^2(1/2A) sec^2(1/2B) 0].](/images/equations/ContactTriangle/Inline14.svg) |
(2)
|
The side lengths of 
are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The area is given by
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
,
, 
, and 
are the area, inradius, semiperimeter, and circumradius, respectively,
of the reference triangle 
. This is the same area as the extouch
triangle.
Beginning with an arbitrary triangle 
, find the contact triangle 
. Then find the contact triangle 
of that triangle, and so on. Then the resulting triangle
approaches an equilateral
triangle (Goldoni 2003). The analogous result also holds for iterative construction
of excentral triangles (Johnson 1929, p. 185;
Goldoni 2003).
The Gergonne point Ge of 
is equivalent to the symmedian
point 
of 
.
The following table gives the centers of the contact triangle in terms of the centers of the reference triangle for Kimberling centers
with 
.
