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

Around Hilbert's theorem: the center of a circle is not constructible by straightedge alone