Exparabolas of a Triangle

TL;DR

This paper explores special exparabolas associated with a triangle, their geometric properties, and how iterative focal triangles converge to equilateral triangles.

Contribution

It characterizes distinguished exparabolas with axes through the centroid and introduces the concept of the $X$-focal triangle with convergence properties.

Findings

Three distinguished exparabolas are determined by a simple cubic equation.

The $X$-focal triangle shares the circumcircle with the original triangle.

Iterated focal triangles converge to equilateral triangles.

Abstract

Among a triangle's exparabolas (parabolas escribed to 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.