Triangles and groups via cevians

Abstract

For a given triangle $T$ and a real number $\rho$ we define Ceva's triangle $\CT_\rho(T)$ 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 $\rho:(1-\rho)$. We identify the smallest interval $\nM_T \subset \nR$ such that the family $\CT_\rho(T), \rho\in \nM_T$, contains all Ceva's triangles up to similarity. We prove that the composition of operators $\CT_\rho, \rho \in \nR$, acting on triangles is governed by a certain group structure on $\nR$. 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 $\CT_\rho$ and $\CT_\xi$ acting on the other triangle.