Antipedal Areas of N-Periodics in the Elliptic Billiard: Row k401 Fails at the Centre and Holds at the Foci

Row \(k_{401}\) of Reznik, Garcia and Koiller's list of experimentally found invariants of \(N\)-periodic billiard trajectories in an ellipse (record AMR-050-0018 of the UnsolvedMath collection) states that for \(N\equiv2\pmod 4\) and every point \(M\) the product \(A'A^*_M\) is constant along the family; here \(A'\) is the signed area of the outer polygon, bounded by the tangents to the ellipse at the vertices, and \(A^*_M\) that of the antipedal polygon of the trajectory with respect to \(M\). The row is false as printed: for the ellipse with semi-axes \(8/5\) and \(1\), two members of the convex \(6\)-periodic family, certified in exact arithmetic, give \(A'A^*_O=801792/21875\) and \(3241/100\) at the centre \(O\), and for every ellipse with semi-axes \(a>b\) the two \(6\)-periodic members symmetric in both axes give values that differ by \(4(a-b)^3(a+b)^5/(ab(2a+b)(a+2b))\). At the foci the row holds. We prove that for every \(N\ge3\) and every member of the convex \(N\)-periodic family \(A^*_{f_1}+A^*_{f_2}=2\kappa A\), where \(A\) is the signed area of the trajectory and \(\kappa\) an explicit rational function of \(a\), \(b\) and the caustic parameter; the proof combines a side-by-side formula for antipedal areas, Stachel's parametrisation by Jacobi elliptic functions and a residue computation. For even \(N\) the two focal areas are equal by central symmetry, so \(A^*_{f_j}=\kappa A\), a factor that an earlier note of the author obtained by another argument, and with Chavez-Caliz's theorem that \(A'A\) is constant for even \(N\) the product \(A'A^*_{f_j}\) is constant; it equals \(16a^2b^2\) for \(N=4\) and \(4a^2(4b^2-a^2)\) for \(N=6\). The vanishing of \(\kappa\) accounts for the families with zero focal antipedal area found earlier, among them \(N=6\) with \(a=2b\), observed by Garcia and Reznik. Non-constancy at points other than the centre and the foci is only tested numerically. No printed statement, proof or refutation of \(k_{401}\) was found in the sources searched. This is an unrefereed note.