On the Fundamentals of Angle Trisectors of a Triangle

Abstract

Over the years of the history of elementary and advanced geometry, trisecting a given angle into three equal parts, was prominent and given more attention. Nevertheless, it is evident that there is a significant research gap of the standard angle trisectors, the lengths of the angle trisectors and the relationships amongst other standard line segments in a triangle. In this paper, we address this gap by developing a purely geometric framework, supplemented with advanced algebraic methods, to obtainclosed-form expressions for internal angle trisectors in a Euclidean triangle. Using the circumcircle, similarity arguments, and Ptolemy’s Theorem, we derive polynomial relations and solve the associated cubic equations explicitly through Cardano’s method. The explicit determination of angle trisector lengths has not been previously available in closed form. Most approaches are trigonometric, but the trisector and related lengths were implicit or incomplete. Moreover, we present few very useful, novel, interesting lemmas, fundamental theorems, and corollaries related to two-dimensional angle trisectors in Euclidean triangles without using any trigonometric, vector algebra or complex number methods.