Vector Proofs: A Quadrilateral thing #2

[forevergone]

I'm not sure if I should've started a new thread for this but..

I need some help trying to prove that the diagonals of a parallelogram bisect each other..
I think I have an idea of how to solve this but I can't seem to put it together:

Given
AB = DC
AD = BC

Known
AB + BC = AC
BC + BD = BD
and so forth..

I'm trying to prove that BZ = ZD and AZ = ZC. Using position vectors, I determined that the midpoint of vector AC to be OA + OB/2 = OZ and that AZ = OA - OZ and ZC = OZ - OC. I had the train of thought in my mind on how to pursue this problem before but I lost it somehow after thinking too hard. I know these are the right steps that need to be considered to finish the problem, but in what steps do I need to do in order to finish this problem?

http://img175.imageshack.us/img175/1889/46wf.th.jpg

 

[lightgrav]

where's the middle of a vector?

 

[forevergone]

its a midpoint. Hence OA + OB/2 = OZ.

So far, my steps are:

1 - AZ = OA - OZ
2 - ZC = OZ - OC

sub into 1 -> AZ = OA - (OA + OB/2)
sub into 2 -> ZC = (OA + OB/2) - OC

but after this, I get lost in trying to prove how AZ = ZC.