The mid-arc triangle is the triangle 
whose vertices consist of the intersections of
the internal angle bisectors with the incircle,
where the points of intersection nearest the vertices are chosen (Kimberling 1998,
p. 160).
It has trilinear vertex matrix
![[(y+z)^2 x^2 x^2; y^2 (x+z)^2 y^2; z^2 z^2 (x+y)^2],](/images/equations/Mid-ArcTriangle/NumberedEquation1.svg) |
where 
,
, and 
.
The incircle is the circumcircle of the mid-arc triangle.
The following table gives the centers of the mid-arc triangle in terms of the centers of the reference triangle for Kimberling centers
with 
.
 | center of mid-arc triangle |  | center of reference triangle |
 | circumcenter |  | incenter |
 | orthocenter |  | first mid-arc point |
 | perspector of abc and orthic-of-orthic triangle |  | third mid-arc point |
