AI 中文总结
本文提出用斜边-角准则、垂线存在性和射线对应原理重构希尔伯特平面中的边角边公理，并推导出边边边和角角边，提供无需连续性假设的更初等公理化。
AI 中文摘要
我们证明，在 $\MG^{-}$ --- $\MG$，即希尔伯特平面，去除边角边公理后 --- 直角三角形的斜边-角判定准则，连同从外部点到直线的垂线的存在性以及一个支配角排序的射线对应原理，足以完整重构边角边公理 --- 因此，作为标准推论，也重构了边边边和角角边。这为希尔伯特本人选择边角边作为公理提供了一个更初等的替代方案，该方案始终限于直角三角形，并且无需任何连续性假设即可证明。最近有几篇论文在类似基础上提出了相关替代方案；我们在结论性评述中将我们的方法与它们进行比较。该结果建立在我们之前关于边角边相关重构的两篇论文之上，并最好与它们一起阅读 \cite{Volpe2026I,Volpe2026II}。
英文摘要
We show that, in $\MG^{-}$ --- $\MG$, the Hilbert plane, deprived of the Side-Angle-Side axiom --- a hypotenuse-angle criterion for right triangles, together with the existence of a perpendicular from an external point to a line and a ray correspondence principle governing the ordering of angles, suffices to reconstruct SAS in full --- and hence, as standard consequences, SSS and SAA as well. This offers a more elementary alternative to Hilbert's own choice of SAS as an axiom, confined throughout to right triangles and provable without any continuity assumption. Several recent papers have proposed related alternatives on comparable grounds; we compare our approach to theirs in the concluding remarks. The result builds on, and is best read alongside, two earlier papers of ours on related reconstructions of SAS \cite{Volpe2026I,Volpe2026II}.
Comments5 pages, 2 fugures
