An Application of Ptolemy's Theorem:Integral triangles with a 120 degree angle and the bisector(of the 120degree angle)also of integral length

TL;DR

This paper characterizes all integral triangles with a 120-degree angle and an integral bisector of that angle, using Ptolemy's theorem and parametric solutions, expanding previous descriptions of such triangles.

Contribution

It provides a new parametric description of integral triangles with a 120-degree angle and an integral bisector, building on prior work with a comprehensive mathematical framework.

Findings

Complete characterization of such triangles using three parameters.

Connection between Ptolemy's theorem and integral triangle properties.

Explicit parametric formulas for the set of triangles.

Abstract

In one of the three 2010/2011 issues of the journal 'MathematicalSpectrum', this author gave a three-parameter description of the entire set of integral triangles(i.e. triangles with integer side lengths)and with a 120 degree angle.This entire set expressed as a union of four families, see reference[5]. In this work we describe, in terms of three parameters again, the set of all integral with a 120 degree angle, and whose bisectors of their 120 degree angles; is also of integral length. To do so, we use the well known historic theorem of Ptolemy for cyclic quadrilaterals, in conjunction with the general positive integer solution of the equation, 1/z=1/x +1/y; and of course in combination with the parametric description of the set of integral triangles with a 120 degree angle mentioned above,The final results of this paper are found in section8.