The geometric mean between positive a and b is 



Multiply both sides by ,

 





Distributing:



Since this must be an identity for all postive a and b

The first term on the left must be identical to the second term on the right
and the second term on the left must be identical to the first term on right.
Let's equate the first term on the left and the second term on the right:



That will be an identity only if the exponents of a are equal and the exponent
of b is 0 since b to the 0 power is 1, that is

 and 


And that is indeed true if  for then we have

 







It's the same when we equate the second term on the left with the first term on right.  

Thus 

Edwin