**Author(s):**Will Murray

**Journal:**Advances in Pure Mathematics

ISSN 2160-0368

**Volume:**03;

**Issue:**01;

**Start page:**116;

**Date:**2013;

Original page

**Keywords:**Abelian Groups; Cubic Equations; Examples; Free Abelian; Geometric Constructions; Group Theory; Integral Triangles; Law of Cosines; Primitive; Pythagorean Angles; Pythagorean Triangles; Pythagorean Triples; Rational Squares | Three-Torsion; Torsion; Torsion-Free; Two-Torsion; Triangle Geometry

**ABSTRACT**

Let 0＜

*γ*＜π be a fixed pythagorean angle. We study the abelian group

*Hr*of primitive integral triangles (

*a,b,c*) for which the angle opposite side

*c*is

*γ*. Addition in

*Hr*is defined by adding the angles

*β*

*opposite side*

*b*and modding out by π-

*γ*. The only

*Hr*for which the structure is known is

*H*

*π*

*/*2, which is free abelian. We prove that for general

*γ*,

*Hr*has an element of order two iff 2(1- cos

*γ*) is a rational square, and it has elements of order three iff the cubic (2cos

*γ*)

*x*3-3

*x*2+1=0 has a rational solution 0＜

*x*＜1. This shows that the set of values of

*γ*for which

*Hr*has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in

*Hr*. Finally, we give some examples of higher order torsion elements in

*Hr*.