Equisectional equivalence of triangles

TL;DR

This paper explores a novel equivalence relation among triangles generated by similarity and a specific edge-division operation, utilizing moduli space and circle properties to characterize equivalence and address rationality and constructibility issues.

Contribution

It introduces a new equivalence relation on triangles, characterizes it using circles of Apollonius within the moduli space framework, and investigates related rationality and constructibility problems.

Findings

Equivalent triangles characterized by Apollonius circles

Established criteria for triangle equivalence in moduli space

Addressed rationality and constructibility issues for special triangles

Abstract

We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of triangles introduced by Nakamura and Oguiso, we give characterization of equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of circles) and properties of special equivalent triangles. We also study rationality problem and constructibility problem.