Geodesic
A case study in Universal Geometry: extending Apollonian circles to relativistic geometry and finite fields
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1227-1257

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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.
Keywords: Apollonian circles, chromogeometry, rational trigonometry, curvature, finite fields. 
@article{SEMR_2020_17_a56,
     author = {J. C. Ho and N. J. Wildberger},
     title = {A case study in {Universal} {Geometry:} extending {Apollonian} circles to relativistic geometry and finite fields},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1227--1257},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a56/}
}
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J. C. Ho; N. J. Wildberger. A case study in Universal Geometry: extending Apollonian circles to relativistic geometry and finite fields. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1227-1257. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a56/