Locus of Intersection for Trisection

TL;DR

This paper presents an algebraic method for angle trisection using the locus of intersection points of two circles, providing a geometric proof equivalent to origami-based trisection.

Contribution

It introduces a novel algebraic locus-based approach for angle trisection, expanding beyond classical Euclidean constraints.

Findings

Derived the locus equation for intersection points of two circles leading to trisection.

Proved the algebraic method aligns with origami-based trisection techniques.

Demonstrated a measurement-free trisection method using geometric loci.

Abstract

Trisecting an angle has been proved to be impossible by Euclidean Geometry, using only straight edge and compass. However, there is a method using Origami (paper folding) procedure to trisect an angle. The algebraic analysis of the same gives us a method of finding trisection using a locus of a point of intersection of two circles. The algebraic analysis and the equation for the locus of the point of intersection of two circles leading to trisection without any measurements is described here. The proof of trisection is exactly same as that of the Origami procedure.