JAMC

Article http://dx.doi.org/10.26855/jamc.2024.06.008

For Arbitrary Angles: A Constructive Approach to Establishing a Consistent Trisection Point Using Compass and Straightedge

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Jiucheng Zhong

Baoxing County Zhongba High School, Baoxing County, Sichuan, China.

*Corresponding author: Jiucheng Zhong

Published: July 15,2024

Abstract

The impossibility of trisecting any angle with a compass and straightedge arises from the inability to trisect the corresponding arc using these tools. This study presents a construction method to identify a fixed trisection point for arbitrary angles. The construction involves drawing a special line segment, which is one-sixth of the chord's length and passes through the midpoint of the chord along with one endpoint. A perpendicular line is then drawn from the other end-point, forming a predefined right-angled triangle. The length of the hypotenuse, plus one-third of the special line segment, becomes a variable segment (Segment 1). The process is iteratively repeated to construct a new right-angled triangle (Triangle 1). This method is further extended to create subsequent right-angled triangles (Triangle 2) by utilizing the hypotenuse of the previous triangle and one-third of a new variable segment (Segment 2). The procedure continues, generating a sequence of variable standard segments. By repeatedly drawing perpendicular lines intersecting the arc, the trisection points of the angle are established.

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How to cite this paper

For Arbitrary Angles: A Constructive Approach to Establishing a Consistent Trisection Point Using Compass and Straightedge

How to cite this paper: Jiucheng Zhong. (2024) For Arbitrary Angles: A Constructive Approach to Establishing a Consistent Trisection Point Using Compass and StraightedgeJournal of Applied Mathematics and Computation8(2), 144-155.

DOI: http://dx.doi.org/10.26855/jamc.2024.06.008