A Web of Confocal Parabolas in a Grid of Hexagons
This work addresses a theoretical geometry problem, presenting incremental findings on known constructions without direct practical applications.
The paper investigates geometric properties arising from constructing regular hexagons on a triangle's sides, revealing surprising relationships including a common isodynamic point, an infinite grid of hexagons and triangles, and a web of confocal parabolas with three foci forming an equilateral triangle.
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.