In ordinary least squares, it can be shown that $\hat\beta_1 = \frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum (x_i - \bar x)^2}$. I'm trying to prove this. I arrived at $$\hat\beta_1=\frac{\overline{xy}-\overline x\cdot \overline y}{\overline{x^2}-\overline{x}^2}$$
Apparently at this point I need to use that $$\overline{(x-\overline x)(y-\overline y)}=\overline{xy}-\overline x\cdot \overline y$$ but I can't seem to prove this. Even for the simplest case $n=2$, if I directly expand both sides of $\overline{(x-\overline x)(y-\overline y)}=\overline{xy}-\overline x\cdot \overline y$, it gets very messy and confusing. Is there an easy way to prove this?
Also, if I only arrived at $\hat\beta_1=\frac{\overline{xy}-\overline x\cdot \overline y}{\overline{x^2}-\overline{x}^2}$ and didn't know that $\overline{(x-\overline x)(y-\overline y)}=\overline{xy}-\overline x\cdot \overline y$ should be true, is there a way to figure out what to do next?