Antreas P Hatzipolakis and Paul Yiu, Reflections in triangle
Forum Geometricorum, 9 (2009) 301--348.
Abstract. This paper is a survey of results on reflections in triangle geometry.
We work with homogeneous barycentric coordinates with reference to a given
triangle ABC and establish various concurrency and perspectivity results related
to triangles formed by reflections, in particular the reflection triangle
P^(a)P^(b)P^(c) of a point P in the sidelines of ABC, and the triangle of
reflections A^(a)B^(b)C^(c) of the vertices of ABC in their respective opposite
sides. We also consider triads of concurrent circles related to these reflections.
In this process, we obtain a number of interesting triangle centers
with relatively simple coordinates. While most of these triangle centers
have been catalogued in Kimberling's Encyclopedia of Triangle Centers
(ETC), there are a few interesting new ones. We give additional properties
of known triangle centers related to reflections, and in a few cases, exhibit
interesting correspondences of cubic curves catalogued in Gibert's Catalogue
of Triangle Cubics (CTC).
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