On Some Geometric Conjectures for Poncelet Polygons

Abstract

In this paper, we generalize some recent observations made by Dan Reznik in a series of computer-assisted experiments. Motivated by the recent proof of one of the original observations, we provide a short proof of a stronger form of the observation by relating it to a classical metric identity involving medians. Our proof is based on a symmetric parametrization for triangles obtained directly from Marden's theorem. This approach does not rely on the assumption that the inconic must be an ellipse contained in the circumcircle. The results apply equally to inellipses whose foci lie outside the circumcircle as well as to inhyperbolas. We also prove a natural extension of Reznik's second observation on odd polygons inscribed in and circumscribed about a pair of homothetic ellipses. Our proof applies equally to crossed odd polygons. Finally, we propose a new conjecture concerning the invariance of the total area of the squares constructed on the sides of cyclic $n$-gons circumscribed about a central conic, thereby placing these area-invariance phenomena within a unified geometric framework.