A case study in Universal Geometry: extending Apollonian circles to relativistic geometry and finite fields

We investigate and extend classical results for the Apollonian circles of a triangle to include relativistic geometries and to hold over general fields, in particular also to finite fields, using the framework of rational trigonometry. Our new results include curvature relations between the three Apollonian circles, criteria for the existence of Isodynamic points, more general formulations of the Lemoine and Brocard axes involving radical axes. Over finite fields the number theoretical aspects of the subject become important.