The triangle 
formed by the points of pairwise intersection of the three intangents. It is not in perspective with 
.
It has trilinear vertex matrix
![[1+cosA cosA-cosC cosA-cosB; cosB-cosC 1+cosB cosB-cosA; cosC-cosB cosC-cosA 1+cosC],](http://mathworld.wolfram.com/images/equations/IntangentsTriangle/NumberedEquation1.svg) | (1) |
or
![[a(b+c-a) (c-a)(c+a-b) (b-a)(b+a-c); (c-b)(c+b-a) b(c+a-b) (a-b)(a+b-c); (b-c)(b+c-a) (a-c)(a+c-b) c(a+b-c)]](http://mathworld.wolfram.com/images/equations/IntangentsTriangle/NumberedEquation2.svg) | (2) |
(Kimberling 1998, p. 161).
It has side lengths
 |  |  | (3) |
 |  |  | (4) |
 |  |  | (5) |
where 
is the inradius of the reference triangle, 
is its semiperimeter, and 
is its circumradius. It has area
 |  |  | (6) |
 |  |  | (7) |
where 
is the area of 
.
The intangents circle is the circumcircle of the intangents triangle.
