Triangles and Groups via CeviansJournal of Geometry
AbstractFor 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 ratioρ: (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.
Citation InformationÁrpád Bényi and Branko Ćurgus. "Triangles and Groups via Cevians" Journal of Geometry Vol. 103 Iss. 3 (2012) p. 375 - 408
Available at: http://works.bepress.com/branko_curgus/8/