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 a²+b² = c², a ≠ b.

(In reply to my attempt by Daniel)

Nice attempt, Daniel! But presumably you could have started at line 6 of your solution, (c-a)^2 >= 0 which is true for all real numbers a and c.

Since the rest of the proof is still valid from that point on, perhaps, Pythagoras wasn't necessary!

 

Posted by JayDeeKay on 2010-03-05 17:59:41