Hi guys. I'm like a few people have been on this post so far: All there bar the three shaded bits. Firstly, everyone, relatively prime means, essentially, if one of the sides has a factor (greater than 1, obviously), then neither of the other two sides have that factor. Another way of saying it is that any two of the sides are coprime. Evidently, if all three of the sides are prime numbers, then it goes without saying it's right. However, notice for the right angles that one of them is of the form 2pq so cannot possibly be prime. (By the way, I didn't use that formula at all in my solving).
Having got that hopefully unambiguous for everyone; I need inspiration as to what is going on in every row and every column. I've tried the following, all of which it isn't:
square numbers, cubes, divisible by 3, products of square numbers, products of cubes. AndrewG-S, dr b, if you are still online - just a tad of a hint of a clue please?