Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them

TL;DR

This paper proves Reznik's conjecture about invariant areas in Poncelet triangles, explores algebraic structures of circle products, and investigates geometric properties of polygons and ellipses formed by these products.

Contribution

It provides a proof for Reznik's conjecture, derives a formula for the invariant sum, and analyzes the geometric and algebraic structures related to these properties.

Findings

Proof of Reznik's conjecture on invariant circle areas

Formula for calculating the total sum of areas

Identification of algebraic structures and geometric properties

Abstract

In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik's conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these products.