The basic arithmetic mean-geometric mean (AM-GM) inequality states simply that if x and y
are nonnegative real numbers, then.
(x+y)/2 ≥ √(x*y), with equality if and
only if x = y.
There are various proofs for this theorem (for any number of values), inter alia Polya, Cauchy, by induction etc.
Now derive your proof directly from Pythagoras' formula a^{2}+b^{2} = c^{2}, a ≠ b.