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 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.

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”.

Language

English

Format

application/pdf

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Included in

Mathematics Commons

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