Triangles and groups via cevians

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