Poncelet Propellers: Invariant Total Blade Area
This work addresses a geometric invariant problem in mathematics, specifically in the study of Poncelet porisms and triangle geometry, which is incremental as it extends known results to more general ellipse configurations.
The paper tackles the problem of finding invariants in the family of Poncelet 3-periodic triangles inscribed in concentric ellipses, showing that the total area of a trio of circumellipses defined by excenters remains constant, even for non-axis-aligned ellipses, and proves an additional invariant involving area ratios to excircles.
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.