A Generalization of Routh's Triangle Theorem

For a given triangle T and a real number ρ we define Ceva’s triangle Cρ(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 MT ⊂ R such that the family Cρ(T ), ρ ∈ MT , contains all Ceva’s triangles up to similarity. We prove that the composition of operators Cρ, ρ ∈ 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 Cρ and Cξ acting on the other triangle.