Mid-Arc Triangle


The mid-arc triangle is the triangle DeltaA^'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 (Kimberling 1998, p. 160).

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.

X_ncenter of mid-arc triangleX_ncenter of reference triangle
X_4orthocenterX_(177)first mid-arc point
X_(24)perspector of abc and orthic-of-orthic triangleX_(2089)third mid-arc point

See also

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

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

Referenced on Wolfram|Alpha

Mid-Arc Triangle

Cite this as:

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

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