The Best Answer still bothers me.
The OP states that the plane flies "at constant airspeed c directly above a closed polygonal path in a plane".
If the desired path over the ground is [east to ] west at a constant "Air speed" and the wind is blowing due north at some velocity, the place will tend to be thrown off course and it will result in some angle [similar to that shown by K-Man above]. As a result the actual distance moved in the E-W direction will be reduced by the resulting angle.
To compensate for this, a pilot must assume a constant "crab angle" in order to stay above the desired line of flight that is choosing a vector that adjusts for the crosswind. The same trig still applies and the "ground speed" will be reduced.
If the wind direction and speed are constant over the entire course throughout the flight, the ground speed will be reduced similarly on some portion of the return flight.
Yeah, I think post #6 should be the Best Answer.