The tangential mid-arc triangle of a reference triangle Δ A B C is the triangle Δ A' B' C' 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. It has trilinear vertex matrix [-y x | z(z + x) | y(y + x) z(z + y) | -z x | x(x + y) y(y + z) | x(x + z) | -x y, ] where x = cos(A/2), y = cos(B/2), and z = cos(C/2). The following table gives the centers of the tangential mid-arc triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=1000.

angle bisector | circumcircle mid-arc triangle | incircle | mid-arc triangle | tangential mid-arc circle