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# Finding m and n if given a 120 degree primitive triple

If given a primitive triangle , we know that one of and is equal to and the other equals where and are relatively prime, positive integers, , and . But, how do we determine which of and equals ?

Lemma 2   If is a primitive integer-sided 120 degree triangle, generated by the parameters and , then if and only if .

Proof. . Then . Since then or for some integer . So, or . Therefore

. Either or . If then . Therefore .

Theorem 6   Let be a given primitive, positive integer-sided triangle. Assign labels and such that ; from Lemma (2), this can be done in exactly one way. Then the fractions

and

(each fraction reduced to lowest terms) is equal to where

and

Proof. Since then . So,

 and

Theorem 7   Let be a given primitive, positive integer-sided triangle. Assign labels and such that ; from Lemma (2), this can be done in exactly one way. Then the fractions

and

(each fraction reduced to lowest terms) is equal to where

and

Proof. Since there exists relatively prime, positive integers and , , such that

and

Then

Let and . Then

and

Hence, we have,

 and

Examples

is a primitive 120 degree triangle. . Therefore . So,

Hence , , and .

is primitive. . Therefore . Thus

Hence , , and .

is a primitive 120 degree triangle. . Therefore . So,

Hence

and

is primitive. . Therefore . Thus

Hence

and

Next: 120 degree and 60 Up: Pythagorean Triples, etc. Previous: Finding parametric equations for   Contents
f. barnes 2008-04-29
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