Conservations of "MacBeath" Poncelet Polygons: sum of squared sidelengths and double-angle cosines

Let P be a family of Poncelet polygons intescribed between an outer circle C and a nested elliptic caustic with one focus on the center of C. We call these "MacBeath" family, since for N=3 the caustic is known as the MacBeath inconic (foci on X3 and X4) [1]. The video shows that for all N, these polygons conserve (i) the sum of squared sidelengths and (ii) the sum of double-angle cosines.
For Liouville-based proof methods see [2,3].
[1] E. Weisstein, "MacBeath Inconic", Wolfram's MathWorld, 2003. mathworld.wolf...
[2] Pedro Roitman et. al, "New Invariants of Poncelet-Jacobi Bicentric Polygons", Arnold Math. J., vol. 7, 2021, pages 619-637.
[3] Arsenyi Akopyan et al., "Billiards in ellipses revisited", European Journal of Mathematics, vol. 8, no. 4. Springer Nature, pp. 1313-1327, 2022.

[@potatoes7825]

Hi, the reason the sum of squares of side lengths of the quadrilaterals is constant 8 is because the condition is equivalent to the circumcenter having an isogonal conjugate with respect to the quadrilateral, which is equivalent to

[@lgooch]

admits

[@dreznik]

brilliant! question, how do u do isog. conj. wrt a quadrilateral? please send me an email? dreznik --at-- g m a i l