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PreprintCC BY-NC-ND 4.0arXiv

On the Reconstruction of SAS from Other Triangle Congruence Criteria, Part II: Eliminating the Pons Asinorum

  • Roberto Volpe
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Abstract

In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [\textbf{RCT}], the existence of angle bisectors [\textbf{AB}], the congruence of supplements of congruent angles [\textbf{SA}], and the Pons Asinorum [\textbf{PA}], suffices to reconstruct SAS. We left open the question of whether [\textbf{PA}] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that SAA, [RCT], [AB] ⊢ [PA], so that [\textbf{PA}] is redundant among the hypotheses of our main theorem, which improves to SAA, [RCT], [AB], [SA] ⊢ SAS. The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.

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Cite

BibTeXRIS
BibTeX
@misc{volpe2026reconstruction,
  title = {{On the Reconstruction of SAS from Other Triangle Congruence Criteria, Part II: Eliminating the Pons Asinorum}},
  author = {Roberto Volpe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21385},
  eprint = {2609.21385},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21385},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the Reconstruction of SAS from Other Triangle Congruence Criteria, Part II: Eliminating the Pons Asinorum
AU  - Roberto Volpe
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21385
UR  - https://arxiv.org/abs/2609.21385
AB  - In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [\textbf{RCT}], the existence of angle bisectors [\textbf{AB}], the congruence of supplements of congruent angles [\textbf{SA}], and the Pons Asinorum [\textbf{PA}], suffices to reconstruct SAS. We left open the question of whether [\textbf{PA}] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that SAA, [RCT], [AB] ⊢ [PA], so that [\textbf{PA}] is redundant among the hypotheses of our main theorem, which improves to SAA, [RCT], [AB], [SA] ⊢ SAS. The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.
ER  -