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

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.