The mid-arc triangle is the triangle Δ A' B' C' whose vertices consist of 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 + z)^2 | x^2 | x^2 y^2 | (x + z)^2 | y^2 z^2 | z^2 | (x + y)^2], where x = cos(A/2), y = cos(B/2), and z = cos(C/2). 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 X_n with n<=100.

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