Without the geometry, the most simple way is this. Let b^2 be the square of the smallest possible natural b, such that it is twice as big as another square of a natural, a^2!
Now, if the b/2 is a natural, even a smaller pair of naturals satisfies the initial condition. And b must indeed be even, since b^2 is even.
(1) is a direct consequence, or another form of: a/b=SQRT(2).
This one has been known since long ago, nothing new here in this post. Just a small clarification of the previous, much more ambitious post.
Done with that.