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

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.