You need a separate proof, but the argument can easily be reversed. Start with a generic circle x^(2) + y^(2) + A'x + B'y + D' = 0 in the plane, then define A,B,C,D such that 2A = A', 2B = B', C + D = 1, D - C = D'; then by the argument in the document the corresponding set on the sphere is its intersection with the plane Ax + By + Ch + D = 0.