From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Triples for 120 degree triangles.
Date: 25 Feb 1999 07:11:18 GMT
Newsgroups: sci.math
Keywords: triangles with integer length sides

Quentin Grady <quentin@inhb.co.nz> wrote:
>Are there triples of integers for 120 degree angle triangles just as
>there are primitive pythagorean triples?

Sure. By the law of cosines what you're asking for is triples  [a,b,c]
of integers with  a^2=b^2+c^2+b*c. Up to scaling, all solutions are
found by [a,b,c]=[x^2+xy+y^2,(x^2-y^2),(2x+y)y]  for some relatively
prime pair of integers  x,y. You probably want  b  and  c  positive and
by symmetry can insist  b > c, which can be accomplished by restricting
the choices for  x  and  y. (You might as well assume  y>0, say, since
(-x,-y) leads to the same  [a,b,c]  as  (x,y).) Your first few primitive
solutions are then  [7, 5, 3], [13, 8, 7], [31, 24, 11], ...

As soon as you observe you're just trying to solve a Diophantine equation,
you can turn to, say,
	index/11DXX.html
	
dave