I've always seen it stated that a Pythagorean Triple is threeintegers $\displaystyle (a, b, c)$ where $\displaystyle a^2+b^2=c^2$.positive

Why the restriction that $\displaystyle (a, b, c)$ are positive?

Similarly with Euclid's formula for generating such triples giving the restriction $\displaystyle m>n$ for...

$\displaystyle a=m^2-n^2$

$\displaystyle b=2mn$

$\displaystyle c=m^2+n^2$

If we let $\displaystyle m<n$ we generate the same |absolute| values with -ve $\displaystyle a$

According to the restrictions $\displaystyle (-a, b, c)$ is not a Pythagorean Triple but the result

$\displaystyle a^2+b^2=c^2$

still holds true.

Are the restrictions merely convention or is there another reason?

Thanks