An Affine Approach to Metric Invariants of Poncelet Polygons

We prove that the total area of the power circles associated with an odd p-gon inscribed in and circumscribed about a pair of homothetic ellipses remains invariant throughout the Poncelet family. Our proof is based on a simple affine averaging principle, which also yields several related quadratic invariants, including explicit formulas for the sums of the squared distances from the center to the vertices, to the side midpoints, as well as for the sum of the squared side lengths. As an application, we show that a convex Poncelet pentagon and the corresponding star Poncelet pentagon, both circumscribed about the same inellipse, have equal total power-circle area. These results unify several metric invariants of odd Poncelet polygons within a common affine-geometric framework.