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arXiv · 2410.18863

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

Mehmet Celik·Mathis Duguin·Jia Guo·Dianlun Luo·Kamryn Spinelli·Yunus E. Zeytuncu·Zhuoyu Zhu
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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.

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Mehmet Celik, Mathis Duguin, Jia Guo, Dianlun Luo, Kamryn Spinelli, Yunus E. Zeytuncu, Zhuoyu Zhu. 2024-10-24. Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them. https://arxiv.org/abs/2410.18863

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