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Pythagoras knew it!! (Posted on 2010-03-05) Difficulty: 2 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution | Comment 3 of 5 |

Let ATB be a line segment with
  x = |AT|  and  y = |TB|.
Construct a semicircle with AB as diameter.
Construct line segment CT with C on the 
semicircle and CT perpendicular to AB.
Triangle ABC is clearly a right triangle
with |CT| <= |AB|/2 ( with equality when
x = y ).
   |CT|^2 = |AC|^2 - |AT|^2
   |CT|^2 = |BC|^2 - |TB|^2
  2|CT|^2 = (|AC|^2 + |BC|^2) - |AT|^2 - |TB|^2
          = |AB|^2 - |AT|^2 - |TB|^2
          = (|AT| + |TB|)^2 - |AT|^2 - |TB|^2
          = 2|AT||TB|
          = 2xy
Therefore,
  sqrt(xy) = |CT| <= |AB|/2 = (|AT| + |TB|)/2 = (x + y)/2
 

  Posted by Bractals on 2010-03-05 18:11:49
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