DOI: https://doi.org/10.36347/sjpms.2025.v12i09.003

Pages: 419-431

There are three classical problems in ancient Greek mathematics that were highly influential in the development of geometry: squaring the circle, trisecting an angle, and doubling the cube. The problem of angle trisection, in particular, involves constructing an angle that is exactly one-third of a given arbitrary angle using only two tools: an unmarked straightedge and a compass. This thesis focuses on the problem of trisecting an arbitrary angle. I present a classical straightedge-and-compass construction that achieves exact trisection, avoiding the explicit use of π and employing a ruler-based geometric analysis and synthesis approach. While it is relatively straightforward to trisect certain special angles (e.g., a right angle), trisecting a general angle has historically been considered impossible under classical constraints. The origins of the trisection problem are difficult to date precisely. The result of this research provides an exact construction-based solution to the long-standing challenge of trisecting an angle using only a straightedge and compass in Euclidean geometry. A solution to this classical problem of trisecting an angle—a challenge that has persisted since ancient Greece—was published in the journal SJPMS [9]. Although that solution employs only theorems and corollaries from high school geometry, it remains somewhat complex. This article, titled "An Exact and Simple Solution to the 'Trisecting an Angle' Problem Using Straightedge and Compass" presents a shorter and simpler solution compared to the 2024 publication. Using the methods of analysis and synthesis, along with straightedge and compass constructions in Euclidean geometry, this paper arrives at an exact—not approximate—solution. Additionally, the study introduces a newly invented tool called the "Trisection Ruler," which enables quick and precise trisection of any given angle.