#### Document Type

Article

#### Publication Date

2012

#### Keywords

Brocard angle, Median triangle, Generalized median triangle, Cevian, Left-circulant matrix, Reflection matrix

#### Abstract

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.

#### Publication Title

Journal of Geometry

#### Volume

103

#### Issue

3

#### First Page

375

#### Last Page

408

#### 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”.

#### Recommended Citation

Bényi, Árpád and Ćurgus, Branko, "Triangles and Groups via Cevians" (2012). *Mathematics*. 52.

https://cedar.wwu.edu/math_facpubs/52

#### Language

English

#### Format

application/pdf