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Parable of the Parabola

Vladimir Dragović, Mohammad Hassan Murad

TL;DR

The paper analyzes triangles and quadrilaterals inscribed in a circle and circumscribed about a parabola, proving a planimetric version of Poncelet's theorem without Cayley-type elliptic conditions. It develops and applies Joachimsthal's tangent notation to parabolas, derives explicit tangent and common-tangent criteria, and establishes precise geometric and algebraic conditions for the existence of n-Poncelet configurations with a circle and a parabola. Key results include a triangle criterion tied to the circle containing the parabola's focus, and antiparallelquadric structures (Darboux butterflies) in the special case where center equals focus, along with a complete description of the 4-Poncelet pairs when the center and focus differ. The work culminates in isoperiodic classifications for n=3 and n=4, providing necessary and sufficient conditions for confocal and nonconfocal families and highlighting a purely geometric, self-contained approach independent of elliptic-curve machinery.

Abstract

We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove it along the way. Similarly, our arguments here \emph{are logically independent} from Cayley's condition, describing points of a finite order on an elliptic curve or any other use of the theory of elliptic curves. Instead, we use purely planimetric methods, including the Joachimsthal notation, to fully describe such polygons and associated circles and parabolas. We prove that a circle contains the focus of a parabola if and only if there is a triangle inscribed in the circle and circumscribed about the parabola. We prove that if the center of a circle coincides with the focus of a parabola, then there exists a quadrilateral inscribed in the circle and circumscribed about the parabola. We further prove that the quadrilaterals obtained in such a way are antiparallelograms. If the center of a circle does not coincide with the focus of a parabola, then a quadrilateral inscribed in the circle and circumscribed about the parabola exists if and only if the directrix of the parabola contains the point of intersection of the polar of the focus with respect to the circle with the line determined by the center and the focus. That point coincides with the intersection of the diagonals of any quadrilateral inscribed in the circle and circumscribed about the parabola. In particular, for a given circle and a confocal pencil of parabolas with the focus different than the center of the circle, there is a unique parabola for which there exists a quadrilateral circumscribed about it and inscribed in the circle.

Parable of the Parabola

TL;DR

The paper analyzes triangles and quadrilaterals inscribed in a circle and circumscribed about a parabola, proving a planimetric version of Poncelet's theorem without Cayley-type elliptic conditions. It develops and applies Joachimsthal's tangent notation to parabolas, derives explicit tangent and common-tangent criteria, and establishes precise geometric and algebraic conditions for the existence of n-Poncelet configurations with a circle and a parabola. Key results include a triangle criterion tied to the circle containing the parabola's focus, and antiparallelquadric structures (Darboux butterflies) in the special case where center equals focus, along with a complete description of the 4-Poncelet pairs when the center and focus differ. The work culminates in isoperiodic classifications for n=3 and n=4, providing necessary and sufficient conditions for confocal and nonconfocal families and highlighting a purely geometric, self-contained approach independent of elliptic-curve machinery.

Abstract

We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove it along the way. Similarly, our arguments here \emph{are logically independent} from Cayley's condition, describing points of a finite order on an elliptic curve or any other use of the theory of elliptic curves. Instead, we use purely planimetric methods, including the Joachimsthal notation, to fully describe such polygons and associated circles and parabolas. We prove that a circle contains the focus of a parabola if and only if there is a triangle inscribed in the circle and circumscribed about the parabola. We prove that if the center of a circle coincides with the focus of a parabola, then there exists a quadrilateral inscribed in the circle and circumscribed about the parabola. We further prove that the quadrilaterals obtained in such a way are antiparallelograms. If the center of a circle does not coincide with the focus of a parabola, then a quadrilateral inscribed in the circle and circumscribed about the parabola exists if and only if the directrix of the parabola contains the point of intersection of the polar of the focus with respect to the circle with the line determined by the center and the focus. That point coincides with the intersection of the diagonals of any quadrilateral inscribed in the circle and circumscribed about the parabola. In particular, for a given circle and a confocal pencil of parabolas with the focus different than the center of the circle, there is a unique parabola for which there exists a quadrilateral circumscribed about it and inscribed in the circle.

Paper Structure

This paper contains 9 sections, 35 theorems, 53 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Joachimsthal's Notation
  3. Joachimsthal's Notation and Parabola
  4. Common Tangents
  5. Triangles inscribed in a circle circumscribed about a parabola
  6. Quadrilaterals inscribed in a circle circumscribed about a parabola
  7. The focus of the parabola coincides with the center of the circle
  8. The focus of the parabola differs from the center of the circle
  9. Isoperiodic families of parabolas with a circle for $n=3$ and $n=4$

Key Result

Lemma 1.1

Let $X$ be an arbitrary point on a parabola $\mathcal{P}$ with the focus $F$ and the directrix $\ell$. Then the tangent to the parabola $\mathcal{P}$ at $X$ is the bisector of the angle between $FX$ and the perpendicular from $X$ to $\ell$. See Figure fig.2.

Figures (18)

  • Figure 1: Illustrations of Poncelet's Theorem.
  • Figure 2: The defining and focal properties of the parabola with the focus $F$ and the directrix $\ell$: $|XA'|=|XF|$ and $\angle A'XA=\angle AXF$.
  • Figure 3: Joachimsthal's section.
  • Figure 4: The pair of tangents from $A$ to $\mathcal{P}$.
  • Figure 5: Points of common tangents at the circle either lie on a hyperbola or pair of straight lines.
  • ...and 13 more figures

Theorems & Definitions (76)

  • Definition 1.1
  • Lemma 1.1: Focal property of parabola
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.1
  • ...and 66 more