Central Symmetry and Seven Area Identities for Even N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many experimentally found invariants of the families of \(N\)-periodic billiard trajectories in an ellipse. Seven of their rows, \(k_{406,a}\), \(k_{607}\), \(k_{609}\), \(k_{610}\), \(k_{905}\), \(k_{906}\) and \(k_{907,b}\) (records AMR-050-0024, -0035, -0037, -0038, -0064, -0065 and -0067 of the UnsolvedMath collection), state for even \(N\) that a ratio of two signed areas, built from the trajectory, its outer or inner polygon and the foci, equals \(1\), or that two centroids lie at the centre. We prove them in the form in which they are true: for every trajectory of even least period whose caustic is a confocal ellipse, convex or self-intersecting, the two signed areas are equal at every member of the family (for \(k_{610}\), at every member where the polygons are defined), and for \(k_{406,a}\) the vertex centroid is the centre and the area centroid is the centre wherever the area is nonzero; so each ratio equals \(1\) wherever its denominator is nonzero. The argument is the one the source used for sibling rows: an even periodic trajectory is centrally symmetric, a classical fact printed by Stachel, and every construction involved commutes with the half-turn and with cyclic relabelling. We give a self-contained proof and record where the ratios are undefined. For \(k_{607}\) the common area vanishes along whole families, also inside the row's class \(N\equiv0\pmod4\): for \(N=8,12,16\) at \(b/a=0.5394187523\ldots\), \(0.5618074057\ldots\), \(0.5688313760\ldots\), for \(N=10\) at \(b/a=0.5543486017\ldots\) and for \(N=6\) at \(b/a=\frac12\). The common areas of \(k_{905}\), \(k_{610}\) and, in star families, \(k_{906}\) vanish at isolated members, the polygons of \(k_{610}\) degenerate exactly at the members with a vertex on a directrix of the caustic, and the area centroid of \(k_{406,a}\) is undefined at isolated members. Apart from this degeneracy criterion these findings are numerical, and nonvanishing elsewhere is not proved. Row \(k_{907,b}\) uses a symbol that the source does not define; the theorem holds for every reading that applies one isometry-natural recipe at both foci. This is an unrefereed note.