Rational trigonometry gives a foundation for a purely algebraic approach to metrical geometry. A big advantage is that the theory works essentially with any bilinear form, not just the Euclidean one, since the usual notions of distance and angle are replaced by the more general notions of quadrance and spread.
In this talk, which is joint work with Nguyen Le, we show how transforming bilinear forms with a linear transformation allows a new framework for the venerable subject of Triangle Geometry, and extends its theorems to relativistic geometries and beyond. We are particularly interested in the Incenter hierarchy, whose natural four-fold symmetry supports remarkable new strong concurrencies, and number theoretic considerations come into play.
This yields quite a few new and surprising facts to be added to Kimberling's monumental Online Encyclopedia of Triangle Centers. Many of our results will be illustrated with triangles from a relativistic setting, showing that this geometry is every bit as interesting as our more familiar Euclidean one.