For all real numbers a and b, 2a•b=a^2 + b^2

That's not true at all.

If  a=2  and  b=3 . . .  then

             2 · a · b = 12
but
            a²  +  b²  =  13 .

This doesn't satisfy your statement, because 12 and 13 are not equal.

Right off hand, it looks to me as if your statement is true
ONLY if 
                   |a|  =  |b|                 ('a' = plus or minus 'b') 

and NOT for any other pairs of numbers.