Tangential Mid-Arc Triangle


The tangential mid-arc triangle of a reference triangle DeltaABC is the triangle DeltaA^'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 (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,]

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.

The tangential mid-arc triangle is perspective with the original triangle, the perspector being Kimberling center X_(177). This perspector is the incenter of the contact triangle of DeltaABC, and is called the first mid-arc point of DeltaABC.

The circumcircle of the tangential mid-arc triangle is the tangential mid-arc circle.

See also

Angle Bisector, Circumcircle Mid-Arc Triangle, Incircle, Mid-Arc Triangle, Tangential Mid-Arc Circle

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Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Tangential Mid-Arc Triangle

Cite this as:

Weisstein, Eric W. "Tangential Mid-Arc Triangle." From MathWorld--A Wolfram Web Resource.

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