preprints.org > computer science and mathematics > geometry and topology > doi: 10.20944/preprints202407.2589.v1 (registering DOI)

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 31 July 2024 / Approved: 31 July 2024 / Online: 1 August 2024 (05:37:38 CEST)

This paper explores the geometric impossibility of angle trisection within Euclidean geometry, with a particular focus on the 90° angle. It presents a rigorous proof that highlights the fundamental limitations of various approaches to angle trisection, highlighting their failure to achieve universality. The paper critiques modern proofs that allow for the trisection of certain angles while dismissing others, arguing that such methods are inherently flawed due to their lack of geometric consistency. The paper introduces a new perspective through a new property named “inconsistent property”, a concept derived from geometric operations involving proportional magnitudes. This property reveals contradictions within the framework of Euclidean geometry, paralleling the logical challenges faced in proving the impossibility of trisecting an arbitrary angle. By drawing comparisons between traditional and contemporary proofs, the paper demonstrates that inconsistencies arise not only from specific cases but also from broader geometric principles. The findings reaffirm the robustness of Euclidean geometry in addressing the trisection problem and challenge the validity of alternative proofs that do not adhere to universal geometric principles. This paper contributes to a clearer understanding of the limitations of angle trisection methods and encourages further investigation into the technical implications and broader impact of the angle impossibility proof.

Angle Trisection; Euclidean Geometry; Impossibility Proof; Geometric Inconsistency; Inconsistent Property; Proof Validity; Geometric Reasoning; Incommensurability

Computer Science and Mathematics, Geometry and Topology

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