Brocard angle, Median triangle, Generalized median triangle, Cevian, Left-circulant matrix, Reflection matrix
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 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.
Journal of Geometry
Required Publisher's Statement
© Springer International Publishing AG
“The final publication is available at Springer via http://dx.doi.org/10.1007/s00022-013-0142-x”.
Bényi, Árpád and Ćurgus, Branko, "Triangles and Groups via Cevians" (2012). Mathematics. 52.