November 19th, 2009, 03:09 PM  #1 
Newbie Joined: Nov 2009 Posts: 12 Thanks: 0  law of cosines
Hello everybody, my question is about pythagorean and "similar"triples. pythagorean triples are triples that confirm the equation . they can be represented in parametres by this form: with m>n, and now its my task to find rules for triples in a triangle with integer sidelengths and a angle of 60° oder 120°. with the law of cosines follows: or. do you have some ideas how to find a presentation of parametres? perhaps with an square addition?(is this the right word in english?) one solution for a triangle with a 120°angle is c=7, a=5 and b=3 and i am looking for a formula to generate all these kind of triples. thanks marten 
November 20th, 2009, 09:36 PM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: law of cosines
Assume that one of the shorter sides is a rational multiple of the sum of the other two. Substitute that into c˛=a˛+b˛+ab and see where it takes you.

November 23rd, 2009, 01:56 PM  #3 
Newbie Joined: Nov 2009 Posts: 12 Thanks: 0  Re: law of cosines
hi aswoods, thank your for your advice, i dont know if i understood you in the right way, is this what you mean? substitution: with when you use this method for triangle with 90°, you get the formula for the unique circle (). but what is ? an ellipse? 
November 24th, 2009, 10:15 PM  #4  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: law of cosines Quote:
We look at , which is less than zero in this case. So it's an ellipse. You could rotate the axis, if desired, but I'm not sure how that would help.  

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