Triangles and groups via cevians

For a given triangle $T$ and a real number $\rho$ we define Ceva’s triangle ${\text{CT}}_{\rho }(T)$ 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 ${\text{nM}}_T\subset \text{nR}$ such that the family ${\text{CT}}_{\rho }(T),\rho \in {\text{nM}}_T$, contains all Ceva’s triangles up to similarity. We prove that the composition of operators $\CT_\rho, \rho \in \nR$,actingontrianglesisgovernedbyacertaingroupstructureon$\nR$. 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 ${\text{CT}}_{\rho }$ and ${\text{CT}}_{\xi }$ acting on the other triangle.