Kill this wives tale

[raburgeson]

Visualize yourself between 2 planets were you seems your mass is equally attracted (so you've been taught) so you float between them. Now let's hold that thought and move back some (or just make it smaller). Now take a third planet and put it in various orbits around this planet pair. You will notice now you are not moving because you are standing in the center of gravity. Not because your mass is equally attracted.

[oouthere]

I'm not certain where you are running with this but you'd be moving toward the third planet.  The two planets cancel each other out, so the only out-of-balance force is the third planet.  Of course, once the disruption happened then you'd be influenced more by which of the original two that you are most drawn toward and then you can practice your dynamic geometry.

Rich

[magnetoelastic]

Since the discovery of Newton's theory of gravitation, it has been demonstrated mathematically that three gravitational bodies cannot be made to exist in stable gravitational orbits.  The gravitational fields will interact, ultimately modifying the eccentricity of orbits, so that there is no stable orbits can be attained.  The sole exception to this paradox is the condition where the three bodies lie at the apexes of an equilateral triangle.  (Interestingly enough, there are minor asteroids that share Earth's orbit, at +/- 60 degrees from the sun, fulfilling this condition.)

For a multi-body system, the most stable case exists where the orbital spacing of successive planets form a harmonic series, where the inter-planetary gravitational potential energy is a minimum.  Not coincidentally, this condition is described by Bode's Law, which predicts the planetary orbits for our Solar System.

[raburgeson]

Yes, the 3rd planet would pull you out of rest. The point is the third planet shows you, your center of gravity was at the center of gravity of the planet pair. If you feel the need remove yourself before you put the third planet into orbit. Don't side track youself with details right now. I'm just trying to point out there is a collective point of gravity between 2 objects. When all of the planets were lined up was the center of gravity for the solar system outside the sun? Could it be that the best place to analyze gravity would be at the center of gravity for the Earth and The Moon? Some say they think gravity is instantaneous, as no mass exists without a complete gravitational field in place it is already present and acts instantly. This does not mean it travels instantly.

Don't let this sway you, but you reservations about stababiliy are real. If you build a craft that has negitive gravity, or it's own field stronger than the Earth's or dark energy field, ?(they made anti-gravity a discusting word) then it will be necessary to do a lot of rigorous work to learn how to control it. The craft will not be stable.
I built one of those popular science hovercrafts, we smashed it up several times. There was no resistance, this type of craft will have no resistance in three dimensional space.