A Hypotenuse-Angle Axiomatization of the Hilbert Plane
Abstract
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}.
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BibTeX
@misc{volpe2026hypotenuseangle,
title = {{A Hypotenuse-Angle Axiomatization of the Hilbert Plane}},
author = {Roberto Volpe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35853},
eprint = {2609.35853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35853},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Hypotenuse-Angle Axiomatization of the Hilbert Plane
AU - Roberto Volpe
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.35853
UR - https://arxiv.org/abs/2609.35853
AB - 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}.
ER -