[EE] Wonder if anyone can help

All

I have a sensor in development that has the following resistor network

T1 - - - - R1 - - - - - R2 - - - - - R3 - - - - T2 
                  |            |
                  R4           R5
                  |            |
T3 - - - - R6 - - - - - R7 - - - - - R8 - - - - T4


R1+R2+R3 = always constant (known)
R1 = R6; R2 = R7; R3 = R8
Therefore R6+R7+R8 = also always constant

During sensing the value of R1, R2, R3, R4 and R5 change.

So R1, R2 and R3 vary but always sum to a constant (they sense one thing), R4 and R5 vary independently (they sense separate things).

I can apply a voltage source to any terminal, a ground to any terminal and monitor the voltages at all.  
 
I figure I can apply gnd to T1, V to T2 and measure T3 and T4, then move the V and Gnd around until I have 6 unique V and Gnd combinations and sets of measurements.

I think I can assume the current draw to make the measurement is for these purposes negligibly, thus resistor at each measurement terminal is out of the equation for that setup.

I need a bit of help figuring out how to get the equations I need to determine the resistances from each of the setups and measurements. Any help would be deeply appreciated.

Thank you