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(Semi-)Invariant Curves from Centers of Triangle Families

Klara Mundilova, Oliver Gross

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 Limaçon trisectrices.

(Semi-)Invariant Curves from Centers of Triangle Families

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 Limaçon trisectrices.
Paper Structure (26 sections, 27 theorems, 106 equations, 9 figures, 4 tables)

This paper contains 26 sections, 27 theorems, 106 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Triangle Centers
  5. Definition and preliminaries
  6. Convention
  7. Properties of triangle centers
  8. Traceable triangle centers
  9. Essentially different triangle centers
  10. Collinear triangle centers
  11. Cyclic-affine triangle center combinations
  12. Constant-affine triangle center combinations
  13. ${\bf \Psi}$-Triangle families
  14. Concatenation of ${\bf \Psi}$-triangle families
  15. Aliquot and nedian triangle families
  16. ...and 11 more sections

Key Result

Lemma 1

Two triangle center functions $\psi_0(a,b,c)$ and $\psi_1(a,b,c)$ describe the same triangle center if for some $f(a,b,c) \in \operatorname{Cyc}({\mathcal{T}})$. We will indicate that two triangle center functions $\psi_0$ and $\psi_1$ describe the same triangle center by $\psi_0 \cong \psi_1$.

Figures (9)

  • Figure 1: Illustration of the notation used in this paper, highlighting geometric interpretations of barycentric coordinates (center) and trilinear coordinates (right).
  • Figure 2: Illustration of the space of triangle functions.
  • Figure 3: Illustration of aliquot (left) and nedian (right) triangle families.
  • Figure 4: Illustrations of properties of aliquot and nedian triangle families discussed in Lemma \ref{['prop:families01']}.
  • Figure 5: Illustration of a triangle in a not decomposable triangle family ${\bf \Psi}$.
  • ...and 4 more figures

Theorems & Definitions (70)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Remark 1
  • Definition 3
  • Definition 4
  • Proposition 1
  • Proposition 2
  • Definition 5
  • Lemma 2
  • ...and 60 more