Sándor N. Kiss and Zoltán Kovács, Isogonal conjugacy through a fixed point theorem,
Forum Geometricorum, 16 (2016) 171--178.

Abstract. For all point X in the plane set the affine combination of orthogonal projections with respect to the sides of a fixed triangle, where the coefficients are the absolute barycentric coordinates of the point with respect to the triangle. We prove that this map is affine if and only if X lies not on the circumcircle, and the only fixed point of this map is the isogonal conjugate of the point X.

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