Regression Proof that the point of averages (x,y) lies on the estimated regression line

Question

How do you show that the point of averages (x,y) lies on the estimated regression line?

Answer 1

To get you started: $\bar y = 1/n \sum y_i = 1/n \sum (\hat y_i + \hat \epsilon_i)$ then plug in, how the $\hat y_i$ are estimated by the $x_i$ and you're almost done.

EDIT: since no one replied, here the rest for sake of completeness: the $\hat y_i$ are estimated by $\hat y_i=\hat \beta_0 + \hat \beta_1 x_{i1} + \ldots + \hat \beta_n x_{in} + \hat \epsilon_i$, so you get $\bar y = 1/n \sum \hat \beta_0 + \hat \beta_1 x_{i1} + \ldots + \hat \beta_n x_{in}$ (the $\hat \epsilon_i$ sum to zero) and finally:
$\bar y = \hat \beta_0 + \hat \beta_1 \bar x_{1} + \ldots + \hat \beta_n \bar x_{n}$. And that's it: The regression line goes through the point $(\bar x, \bar y)$