Why can't this mixer be built with two resistors?
Think about it - what happens when there isn't a third resistor (to 0 volts) and both input voltages are (say) 10 volts - the output will be 10 volts irrespective of what you have set R1 and R2 to be i.e. it doesn't work without a third resistor. Maybe there are some exceptions but generally, it doesn't work.