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November 19th, 2009, 03:09 PM   #1
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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
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November 20th, 2009, 09:36 PM   #2
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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.
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November 23rd, 2009, 01:56 PM   #3
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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?
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November 24th, 2009, 10:15 PM   #4
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Re: law of cosines

Quote:
Originally Posted by MG85
but what is ?
an ellipse?
Written in the form

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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