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

			(1)
			(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 .