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Cycling along Euler road

Dylan Wyrzykowski

TL;DR

The paper introduces $P_lambda$ points to parametrize the Euler line of $d$-dimensional polygons inscribed in a unit hypersphere and proves that these Euler lines arise inductively from sub-polygons via a central homothety, connecting higher-dimensional geometry to planar cases. Special choices of $\lambda$ recover classical centers, with the Euler line containing the centroid, circumcenter, orthocenter, and nine-point center; the construction shows the Euler line of a single point sprouts higher-dimensional Euler lines. By linking to triangle-center theory through Shinagawa coefficients, the authors derive $\lambda = (u+v)/(3u+v)$ for centers with constant coefficients and report a substantial population of Kimberling centers on the Euler line, including 721 centers known as of 2024-12-30. Specializing to the Encyclopedia of Triangle Centers situates well-known centers within the $P_lambda$ family, providing a dimension-spanning perspective on Euler-line phenomena.

Abstract

We introduce the notion of $P_λ$ points, which canonically parametrize points on the Euler line. This allows us to show that the Euler line of any $d$-dimensional inscribed polygon in Euclidean space arises from the Euler lines of its sub-polygons, beginning from the Euler line of a point in the plane. Furthermore, we situate $P_λ$ points in the literature of modern triangle centers.

Cycling along Euler road

TL;DR

The paper introduces points to parametrize the Euler line of -dimensional polygons inscribed in a unit hypersphere and proves that these Euler lines arise inductively from sub-polygons via a central homothety, connecting higher-dimensional geometry to planar cases. Special choices of recover classical centers, with the Euler line containing the centroid, circumcenter, orthocenter, and nine-point center; the construction shows the Euler line of a single point sprouts higher-dimensional Euler lines. By linking to triangle-center theory through Shinagawa coefficients, the authors derive for centers with constant coefficients and report a substantial population of Kimberling centers on the Euler line, including 721 centers known as of 2024-12-30. Specializing to the Encyclopedia of Triangle Centers situates well-known centers within the family, providing a dimension-spanning perspective on Euler-line phenomena.

Abstract

We introduce the notion of points, which canonically parametrize points on the Euler line. This allows us to show that the Euler line of any -dimensional inscribed polygon in Euclidean space arises from the Euler lines of its sub-polygons, beginning from the Euler line of a point in the plane. Furthermore, we situate points in the literature of modern triangle centers.

Paper Structure

This paper contains 3 sections, 3 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $P_{\lambda}$ points and the Euler line.
  3. In the world of triangle centers

Key Result

Theorem 2.2

Let $A_1, A_2, \ldots, A_n \in \mathbb{S}^{d-1}(O,1)$, with $n > 2, d > 1$. For each $i \in \{1,2,\ldots,n\}$, let $P_{\lambda, i}^{n-1}$ denote the $P_{\lambda}$ point of the sub-polygon $A_1A_2\ldots A_n \setminus A_i$, and let $P_{\lambda}^n$ be the $P_{\lambda}$ point of $A_1A_2\ldots A_n$. Then

Figures (2)

  • Figure 1: Theorem \ref{['themG']} for $d=2, n=5$.
  • Figure 2: The Euler point of a cyclic quadrilateral is a $P_{\lambda}$ point.

Theorems & Definitions (8)

  • Definition 2.1: $P_{\lambda}$ points
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Definition 3.1: Shinagawa Coefficients
  • Lemma 3.2
  • Proposition 3.3: Conversion of Shinagawa coefficients
  • proof