Poncelet Propellers: Invariant Total Blade Area

TL;DR

This paper explores a geometric invariant involving Poncelet 3-periodic triangles inscribed in concentric ellipses, revealing that the total blade area of associated circumellipses remains constant regardless of ellipse orientation.

Contribution

It introduces a novel invariant property of Poncelet 3-periodics, demonstrating the constancy of total blade area and related ratios in a general elliptical setting.

Findings

Total blade area remains invariant under ellipse rotation.

The sum of blade-to-excircle area ratios is also invariant.

The invariants hold for both axis-aligned and rotated ellipse pairs.

Abstract

Given a triangle, a trio of circumellipses can be defined, each centered on an excenter. Over the family of Poncelet 3-periodics (triangles) in a concentric ellipse pair (axis-aligned or not), the trio resembles a rotating propeller, where each "blade" has variable area. Amazingly, their total area is invariant, even when the ellipse pair is not axis-aligned. We also prove a closely-related invariant involving the sum of blade-to-excircle area ratios.