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Generalized Chapple-Euler Relation

Vladimir Dragović, Mohammad Hassan Murad

Abstract

We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where the foci of the conic coincide, the condition reduces to the classical Chapple-Euler relation. We also prove that the sum of the squared sides lengths of a Poncelet triangle is invariant over a family of Poncelet triangles inscribed in a circle and circumscribed about a central conic if and only if the circle is centered either at the center of the conic or at one of the foci of the conic, among several other properties of such triangles that we derive.

Generalized Chapple-Euler Relation

Abstract

We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where the foci of the conic coincide, the condition reduces to the classical Chapple-Euler relation. We also prove that the sum of the squared sides lengths of a Poncelet triangle is invariant over a family of Poncelet triangles inscribed in a circle and circumscribed about a central conic if and only if the circle is centered either at the center of the conic or at one of the foci of the conic, among several other properties of such triangles that we derive.
Paper Structure (5 sections, 29 theorems, 65 equations, 10 figures)

This paper contains 5 sections, 29 theorems, 65 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Triangles inscribed in a circle and circumscribed about a Central Conic
  3. Some Properties of Poncelet triangles
  4. An Ellipse Inscribed in a Triangle Centered at its Circumcenter
  5. Area of Poncelet Triangles

Key Result

Theorem 1.1

Let $\mathcal{C}$ and $\mathcal{D}$ be two circles with radii, $R$ and $r$, respectively, and let $d$ denote the distance between their centers. Then there exists a triangle inscribed in $\mathcal{C}$ and circumscribed about $\mathcal{D}$ if and only if

Figures (10)

  • Figure 1: Illustration of a Porism.
  • Figure 2: Polygonal path $P_0P_1P_2P_3$ is inscribed in $\mathcal{C}$ and circumscribed about $\mathcal{D}$.
  • Figure 3: $\triangle ABC$ circumscribes $\mathcal{D}$ if and only if $C=C'$.
  • Figure 4: Theorem \ref{['thm:2.2']}.
  • Figure 5: Pedal curve \ref{['eq:3.1']}.
  • ...and 5 more figures

Theorems & Definitions (59)

  • Theorem 1.1: Chapple 1746, Landen 1755
  • Definition 1.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.2
  • proof
  • ...and 49 more