THE HYPERBOLIC TRIANGLE INCENTER

Abstract

In the present paper, dedicated to the 30th anniversary of Th.M. Rassias' Stability Theorem, we uncover the incenter of a triangle in the hyperbolic geometry of Bolyai and Lobachevsky. The incenter of a triangle in Euclidean geometry is the point of concurrency of the triangle angle bisectors, and it is commonly determined in terms of its barycentric coordinates with respect to the triangle. The aim of this article is to determine the hyperbolic triangle incenter in terms of the hyperbolic barycentric coordinates that we introduce. Instructively, we employ two models of the hyperbolic geometry of Bolyai and Lobachevsky, the Beltrami-Klein ball model, and the Poincare ball model, which are isomorphic to each other. The geodesics of the Beltrami-Klein ball model are Euclidean segments of straight lines, enabling methods of linear algebra to be employed to calculate points of concurrency of geodesics. While it is convenient to calculate the triangle incenter in the Beltrami-Klein ball model, it is convenient to present the result in the Poincare ball model since this model of hyperbolic geometry is conformal to Euclidean geometry.