Suppose R1+R2 = Ra, R3+R4 = Rc and R5+R6 = Rb
Also suppose Vx is the voltage at node Nx, the point where Ra, Rb and Rc join together.
To solve your problem, consider the currents' balance at Nx: (Va-Vx)/Ra + (Vc-Vx)/Rc = Vx/Rb
Solve for Vx as a function of Vc, then calculate the currents flowing in each branch, and finally replace Ra, Rb and Rc with the original resistor series. The rest should be straightforward.
Have fun!
P.S.: you can apply this method even if Vb is not zero. In facts this method is one of the most important analysis method for linear networks, unfortunately I know its name only in Italian...
Thanks, this is very helpful. I spent some time playing around with this and it solved my problems. You pointed me in the right direction and googling for linear networks and network analysis gave some good hits on wikipedia. I just didn't know the what to search for.