How to derive voltage output formula for a strain gauge?
Think of this (bridge) circuit as two simple potential dividers



R3 and R4 divide the excitation voltage in the ratio     

V1 =  R4/(R3 +R4)  * E

R2 an R1 divide the excitation voltage in the ratio

V2 =  R1/(R2 + R1)  *E

The output voltage, e, is the difference between V1 and V2 (assuming no current is taken by the measuring equipment)