  In order to state the theorem in the title formally and to review its
rigorous proof, we extend and make more precise the
Uspenskiy-Shen-Akopyan-Fedorov model of Euclidean constructions with arbitrary
points; we also introduce formalizations for infinite configurations and for
the projective plane. We exemplify the proof method by simpler and not so well
known results that it is impossible to construct the unit length, or a given
point, by compass and straightedge from nothing by means of classical arbitrary
points. On the other hand we construct any given point by compass and
straightedge from nothing by means of arbitrary points determined by horizontal
segments. We quote a "proof" of Hilbert's theorem from the literature and
explain why it is problematic. We rigorously prove Hilbert's theorem and
present three variants of it, the last one for the projective plane.
