Applications of the identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$ I am reading Euclid's elements
I found the algebraic identity
$ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$
I ponder on usage of this identity for $2$ hours.
but I can't click anything.
$a^2 + b^2 = c^2$ can be used when you want to know the direction between $2$ coordinates.
Any example involving this identity?

If I understand properly $$ab + \left(\frac{a+b}{2} - b\right)^2 =ab+\left(\frac{a-b}{2} \right)^2=\frac{1}{4} \Big(4ab+(a-b)^2\Big)=\frac{1}{4} (a+b)^2=\left(\frac{a+b}{2}\right)^2$$
