A Web of Confocal Parabolas in a Grid of Hexagons

TL;DR

This paper explores a geometric construction involving hexagons and triangles, revealing properties like shared points, iterative grid formation, and a web of confocal parabolas with specific focus points.

Contribution

It introduces a novel geometric configuration linking hexagons, triangles, and confocal parabolas with unique properties and iterative extension.

Findings

Shared isodynamic point with original triangle

Infinite grid of hexagons and triangles formed

Confocal parabola web with three distinct foci

Abstract

If one erects regular hexagons upon the sides of a triangle $T$, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with $T$, (ii) the construction can be extended iteratively, forming an infinite grid of regular hexagons and flank triangles, (iii) a web of confocal parabolas with only three distinct foci interweaves the vertices of hexagons in the grid. Finally, (iv) said foci are the vertices of an equilateral triangle.