A right triangle with side lengths $a,b,c$ where $c$ is the hypotenuse and using integers $m, n $ where $ m > n$, we can find Euclid's Formula.
First, given a right triangle with an hypotenuse $\sqrt{c}$ we can set its side lengths to be m and n, giving $c = m^2 + n^2$. Next, this works for all primitives, which can be proven given the greatest side length is always an odd number telling us that when m and n are even and odd, they will at minimum work for all primitives.
Using this we now can find the integer values of a and b.
$(m^2 + n^2)^2 = c^2$
$m^4 + 2m^2n^2 + n^4 = c^2$
$(m - n)^2 + (2mn)^2 = c^2$
Which results in Euclid's formula:
$(2mn, m^2 - n^2, m^2 + n^2)$
Is this proof valid?