Pythagorean Families

Let $(a,b,c) = v, (a',b',c') = v'$ and $a,a',b,b',c,c'$ be non-negative integers such that $f(x,y,z) = x^{2} + y^{2} - z^{2} = 0$ whenever $f$ is evaluated at $v$ or $v'$ $v$ is said to lie in the same family as $v'$ if $v = nv'$ for $n$ a positive integer or one's reciprocal. The question i ask is how many distinct families of solutions of $f(x,y,z) = 0$ are there.

#NumberTheory

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$