Alternative Proofs for the Side Trisector Lengths Theorem of a Triangle

Abstract

In general, discussions regarding the line for side trisectors are still relatively limited compared to angle trisectors. For angle trisectors, most studies only discuss Morley’s Theorem and its extensions. Meanwhile, for side trisectors, existing papers usually calculate the lengths using both the sides and the angles of the triangle. A common problem is how to find the side trisectors length from its opposite vertex when only the side lengths are known. Furthermore, if the side trisector line is extended to form a tangential excircle, can we determine its radius. In this article, we discuss several alternative proofs to determine the lengths produced by side trisectors in a triangle. The main focus is to derive a formula for the side trisectors length using only the original side lengths and to find the radius of the tangential excircle. These proofs are done simply by using several geometric approaches, such as trigonometry, Stewart’s Theorem, and the Pythagorean Theorem. The result provide a standard formula for the trisector length, which is then used to find the radius of the tangential excircle in the constructed triangle