Triangles and groups via cevians

Abstract

For a given triangle $T$ and a real number $ρ$ we define Ceva's triangle $\CT_ρ(T)$ 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 $\nM_T \subset \nR$ such that the family $\CT_ρ(T), ρ\in \nM_T$, contains all Ceva's triangles up to similarity. We prove that the composition of operators $\CT_ρ, ρ\in \nR$, 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 $\CT_ρ$ and $\CT_ξ$ acting on the other triangle.

33 pages, 17 figures