The incentral triangle 
is the Cevian
triangle of a triangle 
with respect to its incenter 
. It is therefore also the triangle whose
vertices are determined by the intersections of the reference
triangle's angle bisectors with the respective
opposite sides.
Its trilinear vertex matrix is
![[0 1 1; 1 0 1; 1 1 0].](/images/equations/IncentralTriangle/NumberedEquation1.svg) |
(1)
|
It is perspective to every anticevian triangle (Kimberling 1998, p. 157).
It is the cyclocevian triangle with respect to Kimberling center 
.
The side lengths of the incentral triangle are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
and its area is
 |
(5)
|
where 
is the area of the reference triangle.
The circumcircle of the incentral triangle is the incentral circle.
The following table gives the centers of the incentral triangle in terms of the centers of the reference triangle that are Kimberling
centers 
.
 | center of incentral triangle |  | center of reference triangle |
 | triangle centroid |  | bicentric sum of pu(32) |
 | orthocenter |  | orthocenter of the incentral triangle |
