关于从其他三角形全等准则重构SAS的研究
On the Reconstruction of SAS from Other Triangle Congruence Criteria
AI总结：
本研究在希尔伯特平面中移除SAS公理，证明角边角、边边边和边角角准则结合射线对应原理及辅助原理可重构SAS，且角边角路径基础更经济。
AI中文摘要：
从希尔伯特平面出发，并移除边角边（SAS）全等公理，我们研究在何种程度上可以从其余经典三角形全等准则出发，以综合方式重构SAS。我们证明，角边角准则，连同与希尔伯特《几何基础》中定理13相对应的射线对应原理，足以重构SAS。我们进一步证明，边边边和边角角准则也足以重构SAS，但需结合射线对应原理和适当的辅助原理——在第一种情况下需要中点的存在性和直角三角形的斜边-直角边准则，在第二种情况下需要角平分线的存在性、等角补角的全等性以及驴桥定理。尽管这两条路径依赖于不同性质的辅助原理，我们表明，一旦建立了共同的斜边-直角边准则，它们便汇聚于一个单一的最终论证。基于从希尔伯特自身的独立性构造改编而来的显式模型进行的元数学分析，对这些重构进行了补充：它表明，仅凭射线对应原理无法重构任何经典准则，并且驴桥定理和斜边-直角边准则各自独立于其各自重构中所用的其余辅助原理。最终图景并非全等准则的形式化层级，但它确实表明，角边角重构所依据的基础在可证明的意义上比从边边边或边角角获得的重构更为经济。
英文摘要：
Starting from a Hilbert plane and removing the Side-Angle-Side (SAS) congruence axiom, we investigate to what extent SAS can be recovered synthetically from the remaining classical triangle congruence criteria. We show that the Angle-Side-Angle criterion, together with a ray correspondence principle corresponding to Theorem 13 of Hilbert's \emph{Grundlagen der Geometrie}, suffices to reconstruct SAS. We further show that both the Side-Side-Side and the Side-Angle-Angle criteria also suffice, once combined with the ray correspondence principle and suitable auxiliary principles -- the existence of midpoints and a hypotenuse-angle criterion for right triangles in the first case, and the existence of angle bisectors, the congruence of supplements of congruent angles, and the Pons Asinorum in the second. Although the two routes rely on auxiliary principles of different character, we show that they converge on a single final argument once a common hypotenuse-angle criterion is established. A metamathematical analysis, based on an explicit model adapted from Hilbert's own independence construction, complements these reconstructions: it shows that the ray correspondence principle alone cannot reconstruct any of the classical criteria, and that the Pons Asinorum and the hypotenuse-angle criterion are each independent of the remaining auxiliary principles used in their respective reconstructions. The resulting picture is not a formal hierarchy of the congruence criteria, but it does show that the Angle-Side-Angle reconstruction rests on a provably more economical basis than those obtained from Side-Side-Side or Side-Angle-Angle.
