The BCI triangle 
of a triangle 
with incenter 
is defined by letting 
be the center of the incircle
of 
, and similarly defining 
and 
.
The triangles 
and 
are in perspective,
the perspector being the first de
Villiers point, which is Kimberling center 
.
The BCI triangle has trilinear vertex matrix
![[1 1+2cos(1/2C) 1+2cos(1/2B); 1+2cos(1/2C) 1 1+2cos(1/2A); 1+2cos(1/2B) 1+2cos(1/2A) 1].](/images/equations/BCITriangle/NumberedEquation1.svg) |
The following table gives some centers of the BCI triangle in terms of the centers of the reference triangle that correspond to
Kimberling centers 
for 
.
 | center of BCI triangle |  | center of reference triangle |
 | ( , )-harmonic
conjugate of  |  | radical center of the Malfatti circles |
 | inner Vecten point |  | incenter |
