Download Deriving 120 degree and 60 degree triples. triples10.pdf
A 120 degree triple is a solution in positive integers to the equation
If, additionally, , and are pairwise relatively prime then is a primitive 120 degree triple, and is a primitive 120 degree triangle. Clearly, if some integer divides any two of the variables , , and then it divides the remaining variable, therefore can be divided out. Similarly, if is primitive then multiply through by the appropriate integer to get your chosen non-primitive solution. Hence, to find all solutions, it's sufficient to find all primitive solutions.
We will need a preliminary result.
where is some positive integer.
Let be a primitive 120 degree triangle. From the triangle inequality, and since and are each non-zero, the sum of the lengths of any two sides is greater than the length of the remaining side. That is, . Therefore, there exists relatively prime positive integers and , , such that
reduced to lowest terms. From whence
Square both sides of (29), multiply through by , then cancel and rearrange terms to get the result,
There are two cases, and .
|and from (29),||(31)|
Substituting these values for and into (32) yields
Therefore, all 120 degree triples where is considered the same as are given by the parametric equations,
for some positive integer ..
Then and are Eisenstein Integers, and is the conjugate of . Note that , and . Thus
If , and are positive integers such that then is a 60 degree triple. If additionally ,and are pairwise relatively prime then is a primitive 60 degree triple. If is a primitive 60 degree triple then is a primitive 60 degree triangle, and vice versa.
Parametric equations for finding all 60 degree triples can be easily derived from the parametric equations for finding all 120 degree triples. (34)
First note that for any positive integer , is a 60 degree triple. That is, where is the length of each side of an equilateral triangle. Clearly, if is a 60 degree triple where then . So, since we already know how to write down the side lengths of an equilateral triangle, the triples such that need not be included in our derivation.
And in the other direction, .
And in the other direction,
From equation (34) and Claim (7), where and , if , is a 60 degree triple if and only if there exists relatively prime positive integers, and , , and such that
Where is some positive integer.
Figure (1) illustrates Claim (7) where in and , and in and .