Abstract
For a given triangle T and a real number ρ we define Ceva’s triangle \({\mathcal{C}_{\rho}(T)}\) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval \({\mathbb{M}_T \subset \mathbb{R}}\) such that the family \({\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}\), contains all Ceva’s triangles up to similarity. We prove that the composition of operators \({\mathcal{C}_\rho, \rho \in \mathbb{R}}\), acting on triangles is governed by a certain group structure on \({\mathbb{R}}\). We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators \({\mathcal{C}_\rho}\) and \({\mathcal{C}_\xi}\) acting on the other triangle.
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This work is partially supported by a grant from the Simons Foundation (No. 246024 to Árpád Bényi).
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Bényi, Á., Ćurgus, B. Triangles and groups via cevians. J. Geom. 103, 375–408 (2012). https://doi.org/10.1007/s00022-013-0142-x
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DOI: https://doi.org/10.1007/s00022-013-0142-x