On the Study of Hyperbolic Triangles and Circles by Hyperbolic\n Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following\nthe adaptation of barycentric coordinates for use in hyperbolic geometry in\nrecently published books on analytic hyperbolic geometry, known and novel\nresults concerning triangles and circles in the hyperbolic geometry of\nLobachevsky and Bolyai are discovered. Among the novel results are the\nhyperbolic counterparts of important theorems in Euclidean geometry. These are:\n(1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem,\n(iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord\nTheorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we\nprefix a gyro to any term that describes a concept in Euclidean geometry and in\nassociative algebra to mean the analogous concept in hyperbolic geometry and\nnonassociative algebra. Outstanding examples are {\\it gyrogroups} and {\\it\ngyrovector spaces}, and Einstein addition being both {\\it gyrocommutative} and\n{\\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the\nmathematical abstraction of the special relativistic effect known as "Thomas\nprecession".\n