Project Euler - Pythagorean Triplets

I’ve been trying to crank through Project Euler questions over the past week or so. They are pretty challenging at first sight, but I’m finding that after looking at them for a few minutes, patterns start revealing themselves and the problem becomes much clearer. Today’s question is the following:

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a² + b² = c² For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.

Find the product abc