Half-Angle Sines, Steiner Pedal Areas and Evolute Areas of N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many quantities that they found experimentally to be constant along the families of \(N\)-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We treat four rows of their tables (records AMR-050-0002, -0028, -0039 and -0032 of the UnsolvedMath collection) for trajectories of least period \(N\) and every winding number, with signed areas; the source's families are simple, and our statements include them. Let \(A\) and \(A'\) be the areas of the trajectory and of its outer polygon and \(\theta_i\) the angles of the trajectory. Row \(k_{108}\) states that \((A'/A)/\prod_i\sin(\theta_i/2)\) is constant for \(N\equiv2\pmod4\). A. Kriebel showed in a Zenodo deposit, with an exact six-periodic example, that this is false; we prove that the product \((A'/A)\prod_i\sin(\theta_i/2)\) is constant on every family with \(N\equiv2\pmod4\), \(N\ge6\), and that the printed quotient is constant on none. Row \(k_{502}\) states that \(A'/A'_{K'}\) is constant for odd \(N\), where \(A'_{K'}\) is the area of the pedal polygon of the outer polygon with respect to its Steiner curvature centroid; we prove that it equals \(c^2\sigma/(2(\pi_g-\pi_f))\), with three further constants \(\sigma\), \(\pi_f\), \(\pi_g\) of the family, wherever it is defined, while its definedness on every family is not proved. Row \(k_{701}\) states that \(A/A_{ev}\) is constant for \(N>4\), \(A_{ev}\) being the area of the evolute polygon; we prove \(A_{ev}=\nu A\) for every \(N\ge5\) with an explicit constant \(\nu\), so the ratio is constant if \(\nu\ne0\), which is observed numerically but not proved. For \(N=6\), \(A_{ev}=-c^4A/(4a^2b^2)\) for every centrally symmetric hexagon inscribed in the ellipse with semi-axes \(a>b\), \(c^2=a^2-b^2\), and for \(N=4\) the ratio is not constant. Row \(k_{603}\), on focal antipedal distances, was proved by Kriebel in another deposit; we only record a confirmation. The proofs use Stachel's parametrisation by Jacobi elliptic functions, the zeros and poles of the quantities on the period torus, and Liouville's theorem. No printed or deposited proof of the product identity, of row \(k_{502}\) or of row \(k_{701}\) was found in the sources searched. This is an unrefereed note.