围绕希尔伯特定理：一个圆的圆心不能由直尺单独构建
Around Hilbert's theorem: the center of a circle is not constructible by straightedge alone
摘要：
为了形式化地陈述标题中的定理并回顾其严格的证明，我们扩展并使具有任意点的欧几里德构造的Uspenskiy-Shen-Akopyan-Fedorov模型更精确;我们还介绍了无限构形和射影平面的形式化。我们用较简单的、不太为人所知的结果来证明这个证明方法，即用圆规和直尺从无到有，用经典的任意点来构造单位长度或给定点是不可能的。另一方面，我们构造任何给定的点，由指南针和直尺从无到有的水平段确定的任意点的装置。我们从文献中引用希尔伯特定理的“证明”，并解释为什么它是有问题的。我们严格证明希尔伯特定理，并提出了三个变种，最后一个为射影平面。
英文摘要：
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.
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