Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The stationary focus of the Kiepert parabola over a special Poncelet triangle family

Mark Helman, Ronaldo A. Garcia, Dan Reznik

TL;DR

This work shows that the focus of the Kiepert in-parabola, $X_{110}$, remains fixed across a circle-inscribed Poncelet triangle family containing an equilateral triangle, with the fixed location given by $(1/f+1/g)^{-1}$. Using a symmetric complex-parameter description and polar duality with respect to incircles and circumconics, the authors connect this stationarity to a stationary Feuerbach point in the corresponding polar tangential family. They further reveal dynamical-geometric consequences: the Kiepert vertex $X_{3233}$ traces a circle with diameter $X_{110}X_{1511}$, and the tangential family inherits a stationary Feuerbach point linked to $X_{110}$. Overall, the paper highlights invariant centers and elegant locus structures arising in specialized Poncelet porisms and their polar/tangential counterparts.

Abstract

We show that the focus of the Kiepert in-parabola remains stationary over a family of circle-inscribed Poncelet triangles which contain an equilateral triangle.

The stationary focus of the Kiepert parabola over a special Poncelet triangle family

TL;DR

This work shows that the focus of the Kiepert in-parabola, , remains fixed across a circle-inscribed Poncelet triangle family containing an equilateral triangle, with the fixed location given by . Using a symmetric complex-parameter description and polar duality with respect to incircles and circumconics, the authors connect this stationarity to a stationary Feuerbach point in the corresponding polar tangential family. They further reveal dynamical-geometric consequences: the Kiepert vertex traces a circle with diameter , and the tangential family inherits a stationary Feuerbach point linked to . Overall, the paper highlights invariant centers and elegant locus structures arising in specialized Poncelet porisms and their polar/tangential counterparts.

Abstract

We show that the focus of the Kiepert in-parabola remains stationary over a family of circle-inscribed Poncelet triangles which contain an equilateral triangle.

Paper Structure

This paper contains 6 sections, 8 theorems, 3 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Computational proofs
  3. Review: Poncelet's porism
  4. Equilateral containment
  5. Stationary focus of the Kiepert Parabola
  6. Stationary Feuerbach point of the polar

Key Result

Lemma 1

A Poncelet family of triangles inscribed in the unit circle and circumscribing an ellipse with foci $f,g\in\mathbb{C}$ contains an equilateral triangle iff $1/f+1/g$ is on the unit circle $\mathbb{T}$.

Figures (5)

  • Figure 1: The contact family $A'B'C'$ is circle-inscribed and envelops a conic which is the polar image of tangents $\tau$ to $\mathcal{E}$ with respect to the incircle. The point $\tau'$ indicated the instantaneous polarity.
  • Figure 2: $T_o'=A'B'C'$ (brown) is an equilateral for which an inconic (green), centered at $\mathcal{O}$, is chosen, with foci at (complex) $f,g$. $ABC$ is a triangle (blue) in the Poncelet family defined by the circumcircle (black) and the chosen inconic. Over the family, $X_{110}$. the focus of the Kiepert parabola (not shown) is stationary at $(1/f+1/g)^{-1}$, as is its antipode $X_{74}$, i.e., called in etc the 'isogonal conjugate of the Euler infinity point'. https://youtu.be/czK2ZQycq24
  • Figure 3: A $T_o=ABC$ in $\mathcal{T}_o$ (blue) is shown (blue) along with its Kiepert parabola (orange), also an inconic, with focus at $X_{110}$. Over $\mathcal{T}_o$, the locus of its vertex $X_{3233}$ is a circle (dashed orange) whose diameter is $X_{110} X_{1511}$. https://youtu.be/5QK-JzN16tM
  • Figure 4: The construction in \ref{['prop:double-inv-1']}: $K$ is chosen on the circumcircle; $A_{eq}$ is a vertex of the equilateral; $T_o=ABC$ is a Poncelet triangle in $\mathcal{T}_o^*$; $\mathcal{B}$ (dashed green) is the external bisector of $\angle K X_3 A_{eq}$; $\mathcal{B}'$ (dashed red) is its reflection about $X_3 A_{eq}$; over all caustic centers $\mathcal{O}$ on $\mathcal{B}'$, $X_{110}$ is stationary, over $\mathcal{T}_o^*$, at the chosen $K$.
  • Figure 5: The construction of \ref{['prop:l35', 'prop:l35-aeq']}. $X_{110}$ (of $T_o=ABC$) is stationary over Poncelet if $\mathcal{O}$ lies on $\mathcal{L}_{35}$ (dashed red), the perpendicular bisector of $X_3$ and $X_5$. A vertex $A_{eq}$ of the equilateral contained in the family is shown.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Corollary 2
  • proof
  • ...and 2 more