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

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.