The tangential mid-arc triangle of a reference triangle 
is the triangle 
whose sides are the tangents to the incircle
at 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
![[-yx z(z+x) y(y+x); z(z+y) -zx x(x+y); y(y+z) x(x+z) -xy,]](/images/equations/TangentialMid-ArcTriangle/NumberedEquation1.svg) |
where 
,
,
and 
.
The following table gives the centers of the tangential mid-arc triangle in terms of the centers of the reference triangle for
Kimberling centers 
with 
.
 | center of tangential mid-arc triangle |  | center of reference triangle |
 | incenter |  | incenter |
 | Euler infinity point |  | isogonal
conjugate of  |
 | external similitude center of circumcircle and incircle |  | third mid-arc point |
 | orthocenter of the contact triangle |  | first mid-arc point |
 |  -Ceva conjugate of  |  | second mid-arc point of anticomplementary triangle |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | crossdifference
of 
and  |
 | direction of vector  , where  |  | isogonal
conjugate of  |
 | odd ( , 2) infinity point |  | crossdifference
of 
and  |
 | crossdifference
of 
and  |  | isogonal
conjugate of  |
The tangential mid-arc triangle is perspective with the original triangle, the perspector being Kimberling
center 
.
This perspector is the incenter of the contact
triangle of 
, and is called the first mid-arc
point of 
.
The circumcircle of the tangential mid-arc triangle is the tangential mid-arc circle.
