(A) is the A.M. between (a) and (b): [ A = \dfrac{a+b}{2} ]

LHS: [ (a-A)^2 + (A-b)^2 = \left(\dfrac{a-b}{2}\right)^2 + \left(\dfrac{a-b}{2}\right)^2 = 2\cdot\dfrac{(a-b)^2}{4} = \dfrac{1}{2}(a-b)^2 ]

= RHS.

Proved.