In a section about linear regression in
Understanding Statistics in the Behavioral Sciences (7th), Robert Pagano, Thomson, 2004, we find the following equation on page 119.
After an explanation of the notation, we find,
We could then construct 
for each score. If we squared each and summed over all the scores, we would obtain
Obtaining this second equation from the first seems remarkable, but the textbook offered no insights on how one could see this.
Here's one possibility.
If we think of vectors
 "\vec{y}=\left(Y_1, Y_2, \ldots,Y_n \right )")
,
 "\overline{y}=\left(\overline{Y}, \overline{Y}, \ldots,\overline{Y} \right )")
, and
 "\hat{y}=\left(Y_1', Y_2', \ldots,Y_n' \right )")
, then the second equation above is the assertion that
This equation is true whenever the vectors
and
are orthogonal.
But
is precisely the orthogonal projection of
onto the space spanned by
 "\vec{x}=\left(X_1,X_2,\ldots ,X_n)")
and
 "\left(1,1,\ldots ,1)")
(
see my earlier blog), so
is in that space and
is in the orthogonal complement.
