(Semi-)Invariant Curves from Centers of Triangle Families

TL;DR

This paper investigates special triangle centers within certain families, identifying semi-invariant and invariant curves, and explores their geometric properties, including connections to classical trisectrices.

Contribution

It classifies semi-invariant and invariant triangle centers in specific families and links these centers to classical geometric curves like trisectrices.

Findings

Four semi-invariant triangle center families identified

Invariant centers related by similarity transformations

Connections to sheared Maclaurin and Limaçon trisectrices

Abstract

We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an affine transformation. We identify four two-parameter families of triangle centers that are semi-invariant and determine which are invariant, in the sense that the resulting curves for different initial triangles are related by a similarity transformation. We further observe that these centers, when combined with the aliquot triangle family, yield sheared Maclaurin trisectrices, whereas the nedian triangle family yields Lima\c{c}on trisectrices.