Jesus Torres, The triangle of reflections,
Forum Geometricorum, 14 (2014) 265--294.
Abstract. This paper presents some results in triangle geometry discovered with the aids of a dynamic software, namely, the Geometer's Sketchpad, and confirmed with computations using Mathematica 9.0. With the method of barycentric coordinates, we study geometric problems associated with the triangle of reflections T' of a given triangle T (obtained by reflecting the vertices in their opposite sides), resulting in interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. In particular, we show that the Parry reflection point (reflection of circumcenter in the Euler reflection point) is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T' appears as the locus of the perspector of a family of triangles perspective with T', and in a different context as the locus of the orthology center of T' with another family of triangles.
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