Albrecht Hess, Transforming tripolar into barycentric coordinates,
Forum Geometricorum, 15 (2015) 253--261.
Abstract. A simple construction is presented to find a point with given tripolar coordinates, i.e. the ratios of its distances to the points A, B, C of a reference triangle. This construction leads to a very nice transformation formula for tripolar into barycentric coordinates, that simplifies considerably an already existing transformation formula in Kimberling's Encyclopedia of Triangle Centers. The necessary and sufficient conditions for the constructibility are encoded in a triangle whose side lengths are products of the side lengths of ABC with the tripolar coordinates. Formulas for the area of this triangle are presented showing the role of inversion in this construction. As applications, one-line proofs for the formula of the pedal triangle area and the factorization of the dual of the circumcircle are given as well as simplifications of some formulas from ETC.
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