Properties of Parabola-Inscribed Poncelet Polygons

We investigate properties of Poncelet $N$-gon families inscribed in a parabola and circumscribing a focus-centered circle. These can be regarded as the polar images of a bicentric family with respect to the circumcircle, such that the bicentric incircle contains the circumcenter. We derive closure conditions for several $N$ and describe curious Euclidean properties such as straight line, circular, and point, loci, as well as a (perhaps new) conserved quantity.