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 Pythagoras knew it!! (Posted on 2010-03-05)
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 No Rating

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 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 trianglewith |CT| <= |AB|/2 ( with equality whenx = 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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