if A is the AM between a and b prove that
[a] (A+2a)/(A-b) + (A+2b)/(A-a) =4
[b] A-b=(b-a)^2
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A=a+b2.A=a+b2. So
a+b2+2aa+b2−b+a+b2+2ba+b2−aa+b2+2aa+b2−b+a+b2+2ba+b2−a
=5a+ba−b+a+5b−a+b=5a+ba−b+a+5b−a+b
=5a+ba−b−a+5ba−b=5a+ba−b−a+5ba−b
=4a−4ba−b=4a−4ba−b
=4
a+b2+2aa+b2−b+a+b2+2ba+b2−aa+b2+2aa+b2−b+a+b2+2ba+b2−a
=5a+ba−b+a+5b−a+b=5a+ba−b+a+5b−a+b
=5a+ba−b−a+5ba−b=5a+ba−b−a+5ba−b
=4a−4ba−b=4a−4ba−b
=4
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