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Exparabolas of a Triangle

Martin Lukarevski, Hans-Peter Schröcker

TL;DR

The paper investigates exparabolas inscribed in a triangle and their focal geometry, focusing on max-exparabolas whose axes pass through the centroid. It derives a simple cubic equation in t whose real roots yield the three max-exparabolas and proves that their axes concur at the centroid; it then introduces the X-focal triangle (foci of the exparabolas with axes through X), whose orthocenter is X and whose vertices lie on the circumcircle. For X equal to the centroid, the focal-triangle construction yields a dynamical system on successive focal triangles whose even and odd subsequences converge to equilateral triangles, with the limit vertices forming a regular hexagon together with the original focal points. The work intertwines barycentric and Bézier parametrizations to provide explicit cubic conditions and highlights several avenues for further exploration of exparabolas and their focal configurations in triangle geometry.

Abstract

Among a triangle's exparabolas (parabolas inscribed into the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point $X$. Their focal points determine another triangle which we call the $X$-focal triangle. It shares the circumcircle with the original triangle and its orthocenter is $X$. The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles.

Exparabolas of a Triangle

TL;DR

The paper investigates exparabolas inscribed in a triangle and their focal geometry, focusing on max-exparabolas whose axes pass through the centroid. It derives a simple cubic equation in t whose real roots yield the three max-exparabolas and proves that their axes concur at the centroid; it then introduces the X-focal triangle (foci of the exparabolas with axes through X), whose orthocenter is X and whose vertices lie on the circumcircle. For X equal to the centroid, the focal-triangle construction yields a dynamical system on successive focal triangles whose even and odd subsequences converge to equilateral triangles, with the limit vertices forming a regular hexagon together with the original focal points. The work intertwines barycentric and Bézier parametrizations to provide explicit cubic conditions and highlights several avenues for further exploration of exparabolas and their focal configurations in triangle geometry.

Abstract

Among a triangle's exparabolas (parabolas inscribed into the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point . Their focal points determine another triangle which we call the -focal triangle. It shares the circumcircle with the original triangle and its orthocenter is . The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles.

Paper Structure

This paper contains 7 sections, 9 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Parabolas Inscribed into a Triangle
  3. Exparabolas of Maximal Parameter
  4. The Axes of Exparabolas
  5. The Focal Triangle of the Max-exparabolas
  6. Conclusion
  7. Formulas for Bézier Parabolas

Key Result

Lemma 1

For any exparabola there exists $t \in \mathbb{R} \setminus \{0,1\}$ such that its points of tangency with the triangle sides have homogeneous barycentric coordinates cf. Figure fig:inscribed-parabola. (The reason for the seemingly strange labeling of points will become clear later, when we consider three point triples of above type.)

Figures (6)

  • Figure 1: Parabola inscribed into triangle $ABC$, Steiner circumellipse $s$, and focal point $F$ on the circumcircle.
  • Figure 2: Exparabolas of triangle $ABC$.
  • Figure 3: A triangle inscribed into the Steiner circumellipse.
  • Figure 4: Region for admissible points $X$ in Theorem \ref{['th:X-exparabola']}. The axes of the max-exparabolas intersect in the triangle's centroid $G$.
  • Figure 5: The altitudes of the focal triangle $F_AF_BF_C$ are the axes of the max-exparabolas.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1: lukarevski25
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Proposition 1
  • Theorem 2
  • proof
  • ...and 9 more