Prove the equality $(\sum x_i y_i)^2 = (\sum x_i^2\sum y_i^2)$

Question

Can you give an idea of how to prove equality $(\sum x_i y_i)^2 = (\sum x_i^2\sum y_i^2)$ where $Y_i = \alpha + \beta X_i +\epsilon_i, y_i = Y_i -\bar{Y}, x_i = X_i -\bar{X}$ and $ \hat{y_i}= \hat{Y_i} -\bar{Y}$. I just tried to substitute but it didn't work out.

Answer 1

$x_1=1,y_1=0, x_2=0,y_2=1$ gives a counter-example. I will let you choose your parameters so that $x_i$'s and $y_i$'s have these values.

Answer 2

This can't possibly be true. The left hand side multiplied out contains terms like $x_i x_j y_i y_j$, on the right hand side all variables appear squared.