What is the relationship between (x | Z[a]), (y | Z[b]), and (((x^2 + y^2)^(1 / 2)) | Z[a])? In other words, so to speak, how would I create a formula that would solve $$(\sqrt{(x | Z[a])^2 + (y | Z[b])^2}) | Z[a]$$ for either $x$ or $y$, given the other?
In English, what I want is a sequence of coordinates with $x$ as a multiple of $a$, and $y$ as a multiple of $b$, with the length of the line drawn between them also being a multiple of $a$, given that $a$ and $b$ are integers. If this is even possible.
 A: So here's our restatement (see discussion in comments above):
$$
(ma)^2 + (nb)^2 = (pa)^2
$$
Please note that this means that $(ma, nb, pa)$ must form a Pythagorean triple, with hypotenuse $pa$.  So you're trying to find all Pythagorean triples where the hypotenuse and one of the legs have a common divisor, $a$.
http://en.wikipedia.org/wiki/Pythagorean_triple
Since every pair of integers has a common divisor (in some cases the GCD is 1, but it's still a common divisor), you basically just need to find all Pythagorean triples.
Note: if the three sides of a Pythagorean triple are all coprime, then we call it a primitive Pythagorean triple.  If any two sides in a Pythagorean triple have a common divisor $d$, then all three sides share that divisor (why?), so every Pythagorean triple is a multiple of a primitive Pythagorean triple.  Thus you may always take $b=a$ in our equation above, and solutions with $a=1$ correspond to primitive Pythagorean triples.
There are various formulas for generating Pythagorean triples, which would solve your problem completely.
