This expression is always a perfect square How to show that for $x,y\in \Bbb R$,  the  expression $xy+\left(\frac{x-y}{2} \right)^2$ is always a perfect square?  
For example $x=7, y=3$, $7\times 3+\left(\frac{7-3}{2} \right)^2=25=5^2$

$$xy+\left(\frac{x-y}{2} \right)^2\\
=\frac{4xy+x^2+y^2-2xy}{4}\\
=\left(\frac{x+y}{2}\right)^2$$
