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?

You can use it to find all ways of writing $c=a^2-b^2$ with $a$ and $b$ positive natural numbers (or any two real numbers) for fixed $c$. To see this, your identity says:
$$ab=\left ( \frac{a+b}{2} \right )^{2} - \left ( \frac{a-b}{2} \right )^{2}$$
Now pick a number, $c=16$. Then we can factorize $c=2*8$
, so that $a=2$, $b=8$, giving $16=5^{2}-3^{2}$. This should be familiar as $3^{2}+4^{2}=5^{2}$.
We can find all solutions for fixed $c$. This trick relies on unique factorization and the fact that $(1,1)$ and $(1,-1)$ form a basis of $\mathbb{R}^{2}$. With a simple counting argument we give a formula for the number of solutions. If the prime factorization of $c$ is $2^{n_1}p_2^{n_2}...p_k^{n_k}$
$$N=\frac{1}{2}\left ( f(n_1)(n_2+1)(n_3+1)...(n_k+1)-g(c) \right )$$
Where $f(n_1)=n_1-1$ if $n_1>1$, and $1$ if $n_1=0$ and $g(c)=1$ if $c$ is a square, $0$ otherwise.
