The triangle of reflections

This thesis presents some results in triangle geometry discovered using dynamic
software, namely, Geometer’s Sketchpad, and confirmed with computations using
Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated
with the triangle of reflections T of a given triangle T, yielding 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. Particularly, we show that the Parry 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.