Let's start from the pi-network on the input with R3 going to GND (vcc/2 is used because op-amp is supplied from Vcc to GND). By using Thevenin we have:
Voc=R3/(Ra+Rb)*Vi
Rth=Rb+Ra*R3/(Ra+R3)
so all the pi-network can be replaced by a voltage generator Voc followed by a series resistor Rth. The circuit is now a simple inverting-amplifier, thus
Vo=-Rf/Rth*Voc
replacing with complete expression for Voc and Rth we will have:
Vo=-RF*(Ra+R3)/[Rb*(Ra+R3)+Ra*R3] * R3/(Ra+R3) *Vi ==> Vo/Vi=-Rf*R3/[Ra*Rb+(Ra+Rb)*R3]
Since Ra=Rb (let's call this Rs):
Vo/Vi=-Rf*R3/(Rs²+2*R3*Rs) the article says that Rf=2*Rs (have a look at the number written on the components) then:
Vo/Vi=-2*R3/(Rs+2*R3)
normalizing to Rs=1:
Vo/Vi=-2*R3/(1+2*R3)
from this we can now extract R3 normalized (of couse we need tha absolute value of the gain, so the sign "-" have to be eliminated):
(Vo/Vi)+2*R3*(Vo/Vi)=2*R3
from which:
R3=(Vo/Vi)/{2*[1-(Vo/Vi)]}