Proving the SST = SSE + SSR

I need help with this last part of this proof. This is my first time using the forums so I'm not sure how to write all the symbols but I will make it as clear as possible:

SST = SUM [(yi-yavg)^2]

SST = SUM [(yi-yavg + yi hat - yi hat)^2]

This ends up afters foiling:

SST = SSE + SSR + 2SUM[(yi hat - yavg)(yi - y hat)]

I can't figure out how to show that the last term is zero. My professor says to use

y hat = B0+B1*Xi

B0 = yavg -Bi*xavg

and

B1 = SUM [ ((xi - xavg)*(yi-yavg))/((xi-xavg)^2)]

but I can't get it to work out. I get

Which leads me to

SUM[ xi * ( yi - yavg + SUM[ ((xi - xavg)*(yi-yavg))/((xi-xavg)^2)] * (xavg - xi)]

Maybe I am forgeting a trick for dealing with nested summation..

Help! and Thanks in advance!