- Cosmin Pohoata, Geometric constructions of the mixtilinear incircles, Crux Mathematicorum with Mathematical Mayhem, to appear, 2008. (Link to the submitted PDF file)
The term mixtilinear incircles of a triangle was introduced by L. Bankoff [1] naming the three circles each tangent to two sides and to the circumcircle internally. In the same paper, he establishes the “fundamental formula”, as P. Yiu [2] reminds it, expressing the radius of a mixtilinear incircle in terms of the triangle’s inradius. More precisely, consider a triangle ABC and its mixtilinear incircle in the angle A, with center K_{A}, and radius \rho_{A}. Then, r=\rho_{A} \cdot \cos^{2}{\frac{\alpha}{2}}, where r is the inradius of the triangle, and \alpha is the magnitude of the angle at A. In this paper, we present both Bankoff and Yiu’s geometric (ruller-and-compass) constructions of these mixtilinear incircles, and moreover, we shall give three new ones.
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