We investigate and extend classical results for the Apollonian circles of a triangle to include relativistic geometries and to hold over general elds, in particular also to nite elds, 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 nite elds the number theoretical aspects of the subject become important.
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