JAMC
The Geometric Iteration Law of Dividing Any Angle into Three Equal Parts and Its Proof
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Abstract
The classical compass-and-straightedge method for trisecting an arbitrary angle is unsolvable when the corresponding arc is a segment of a circle, as it involves transcendental numbers. This study builds upon classical arguments while acknowledging the limitations of compass-and-straightedge constructions for direct trisecting. By utilizing the chord-arc relationship, we developed a construction method to approximate the one-third chord length of an arc, then iteratively generated segments equal to the chord length. Through extensive exploration, we successfully created two sets of construction diagrams that approximate the one-third chord length, enabling iterative generation of target segments. Using one set of these segments as special cases, we demonstrated that even under the constraint of trisecting the given arc, target segments can be iteratively produced. Since both proofs rely on numerical equality calculations—disregarding rigorous geometric principles—we employed geometric constructions to validate these methods.
Keywords
The corresponding arc to a given angle; the corresponding chord to a given angle; the one-third chord of the arc; the target line segment; constructing a line segment approximating the target line segment
References
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How to cite this paper
The Geometric Iteration Law of Dividing Any Angle into Three Equal Parts and Its Proof
How to cite this paper: Jiucheng Zhong. (2025)The Geometric Iteration Law of Dividing Any Angle into Three Equal Parts and Its Proof. Journal of Applied Mathematics and Computation, 9(4), 306-319.
DOI: http://dx.doi.org/10.26855/jamc.2025.12.010
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