The car overheats in slow traffic because the cooling fan just does not engage.

I have a constant 12 volt reading on the plug going into the fan at all times. There is a little third wire that reads about 2 volts. When the car overheats that wire still reads 2 volts.

Is it correct to assume that the third wire is the control wire that should read 12 volts when the motor gets hot? Is it this wire that turns the motor on?

I have applied 12 volts to the plug and a positive 12 v to the little wire to turn the fan on, but the fan did not run.

Is there a relay somewhere? Is the relay built into the cooling fan assembly? This would require a new cooling fan for $700. Or is there a thermal switch somewhere I need to replace?

Thank you for your help.