Triangles and groups via cevians

TL;DR

This paper explores the properties of Ceva's triangles formed by cevians in a given triangle, revealing a group structure governing their composition and characterizing triangles with the same Brocard angle through iterative transformations.

Contribution

It introduces a group structure on the set of cevian-based triangle transformations and characterizes triangles sharing the same Brocard angle via iterative applications of these operators.

Findings

Identifies the minimal interval of  that generates all Ceva's triangles up to similarity.

Establishes a group structure on the set of cevian transformations.

Characterizes triangles with the same Brocard angle through iterative operator applications.

Abstract

For a given triangle $T$ and a real number $\rho$ we define Ceva's triangle $\CT_\rho(T)$ 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 $\rho:(1-\rho)$. We identify the smallest interval $\nM_T \subset \nR$ such that the family $\CT_\rho(T), \rho\in \nM_T$, contains all Ceva's triangles up to similarity. We prove that the composition of operators $\CT_\rho, \rho \in \nR$, acting on triangles is governed by a certain group structure on $\nR$. 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 $\CT_\rho$ and $\CT_\xi$ acting on the other triangle.