ArticleOpen Access http://dx.doi.org/10.26855/jamc.2025.09.003
A Ruler and Compass Can Be Used to Make a Line Segment Equal to One-third of the Chord of Any Angle
Jiucheng Zhong
Baoxing County Zhongba High School, Ya'an 625000, Sichuan, China.
*Corresponding author:Jiucheng Zhong
Published: October 27, 2025
Abstract
After the publication of "The Construction of a Third-Equal Point on an Arbitrary Angle Using Ruler and Compass" in 2024, the author found that while the construction method and proof were correct, some readers believed that the title, discussion, construction, and proof did not accurately convey the original intent of the construction method. This led to a misunderstanding that the ruler and compass could directly construct a third-equal point on an arc of any angle, thus contradicting the author's claim that this construction method does not contradict the original proof that the ruler and compass cannot trisect an arbitrary angle. To more clearly demonstrate that the construction method can trisect an arbitrary angle without contradicting the original proof that the ruler and compass cannot trisect an arbitrary angle, this paper conducted a very rigorous and serious review of the original work. It was found that both the title and the discussion of the original work failed to express the author's original intent. Therefore, it is necessary to re-prove the essence of the construction method using the title "The Con-struction of a Segment Equal to One-Third of the Chord Length on an Arbitrary Angle Using Ruler and Compass," which is the innovation of this construction method: first, construct a segment equal to one-third of the chord length on the angle, thereby constructing the third-equal point and the third-equal line of the angle, rather than directly constructing a third-equal point on an arc of any angle. In addition to selecting different examples, this paper will also provide example calculations for angles close to 180 degrees and 0 degrees to demonstrate the accuracy and universality of this construction method. This method for trisecting an arbitrary angle can also be used to directly calculate one-third of the chord length on an arbitrary angle, given the side length and chord length of the angle.
Keyword
Arbitrary Angle; Angle Side; Chord; Arc; Arc Height; One-Third of the Angle Chord; Special Line Segment; Isosceles Trapezoid
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Copyright
© 2025 by the author(s).
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives (CC BY-NC-ND) license, which permits non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited and is not modified or adapted.
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How to cite this paper
A Ruler and Compass Can Be Used to Make a Line Segment Equal to One-third of the Chord of Any Angle
How to cite this paper: Jiucheng Zhong. (2025) A Ruler and Compass Can Be Used to Make a Line Segment Equal to One-third of the Chord of Any Angle. Journal of Applied Mathematics and Computation, 9(3), 175-187.
DOI: http://dx.doi.org/10.26855/jamc.2025.09.003