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In hyperbolic triangle geometry, the three classical triangle centers (the barycenter, the circumcenter and the orthocenter) do not align: there is no hyperbolic Euler line. It turns out, however, that there are two exotic analogues of the barycenter (the pseudobarycenter) and the circumcenter (the pseudocircumcenter) that lie on the same line as the orthocenter. This is the Euler-Wildberger line of the triangle. In this paper we provide an analytic proof of this fact, by using barycentric coordinates in the projective model of the hyperbolic plane. In the process, we find the barycentric coordinates of the two exotic centers and we, also, prove that when the curvature of the hyperbolic plane goes to zero, the exotic centers turn into their classical analogues, while the Euler-Wildberger line turns into the Euler line of the limit Euclidean triangle.
| Topic | History of science |
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