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

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.

Figures (3)

• Figure 1: The three circles are separately centered at the midpoint of each side of the Poncelet triangle and pass through the opposite vertex of the triangle.
• Figure 2: The curvature functions of the degree-$3$ Blaschke ellipse and degree-$4$ Blaschke ellipse (blue dashed line and solid black line, respectively).
• Figure 3: Example of Proposition \ref{['prop:reducible-blaschke-hyperbolic-geodesics']} for a degree-$6$ reducible Blaschke product $B$ with $a = 0.2 + 0.3i$.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 2.1: Reznik's conjecture, Reznik
• proof : Proof of Theorem \ref{['thm triangles']}
• Remark 2.2
• Example 3.1
• Remark 3.2
• Example 3.3
• Definition 3.4
• Proposition 3.5
• proof