<Checks> PossiblyDoppelgänger, also known as Journey to the Far Side of the Sun. The idea has also shown up in novels and comic books; the Counter-Earth of Marvel for example.
If it’s at exactly 180 degrees opposite the Earth, it’ll stay there. But if it’s a little bit off of that, it’ll get more off, until eventually the two planets are right next to each other (at which point you start getting some interesting interactions between the two).
Thanks. Doppelgänger is the one I was thinking of. It was mentioned here before, and although I haven’t seen it, it looks like a cheezy old flick I would enjoy, along the lines of Forbidden Planet.
Ah. Got it. Thanks.
So would the 60 degree configuration be more stable than the 180 degree configuration?
Short answer, yes.
All the above answers are wrong. 
You can’t put a second Earth in any of the Lagrange points because it would be unstable due to the influence of the other planets. What you need is another nearby star and have the planet orbit that. One that’s perhaps a light-week away. Close, but not too close.
That’s a fave New Age idea; an invisible twin. However, due to minor perturbations and wobbles from passing stuff, it wouldn’t be hidden permanently. Long before it becomes a serious threat to Earth as Chronos says, it will become visible to us. Our planetary probes would spot it pretty soon after leaving Earth orbit, when the Sun is no longer in the way.
If you could have it tidally locked to the sun the ‘sunlit’ side may be warm enough. Likewise a tidally locked planet that was very close to the sun could have a habitable ring
Not all the answers above were at Lagrange points. Making the extra planet a binary with Earth (replacing the moon) should be stable. The co-orbit idea suggested by ZenBeam[sup]1[/sup] may work, although I suspect you’d need to circularize the Earth’s orbit to make it stable.
Another possible stable place is to put it at 1 AU but with a very high inclination (i.e roughly perpendicular to Earth’s orbit). Time it so that when the new planet crosses the ecliptic, the Earth is on the far side of the Sun.
[sup]1[/sup] Dammit, ZenBeam, that’s my usual addition to these kinds of threads. Stop stealing my ideas 
Then let’s add that mass to Mars.
If added properly (depending on your view and time schedule) it may heat the planet sufficiently to re-melt the core and add a magnetosphere. Then, after a couple of hundred thousand years, you start adding the icy asteroids to build atmosphere and an ocean on the surface.
Gor - Wikipedia is another example. Books, movie, and a whole bunch of nuts who … oh never mind.
Well, it’d drift until it hit a stable Lagrange point. Then it’d stay put fairly well until destabilized by the other planets, as Quartz claims. I’ll take his word for it. My guess is that would take a while.
I like that idea. But wouldn’t that more or less guarantee an eventual collision? I also shudder to think what the other planets would do to its orbit.
Apart from the problem of interactions with other planets. And would you not have to change the solar-orbital period of Earth and its binary?
Actually, it’ll drift past that Lagrange point, then start drifting back the other way until it passes the other one, and so on, looping back and forth.
There are no stable Lagrange points when two of the masses are about the same mass. As Chronos pointed out above, they’re only stable when the third mass is much, much less than the other two.
Yeah, eventually they will fall out of sync, although it would no doubt take a very long time. But they probably won’t crash, at least not immediately. Instead, they will come close enough to affect each other’s orbit. At which point the orbits will probably become unstable. They could eventually crash (either with each other or with some other planet) some time after that or be ejected from the system.
What’s needed is to put the two planets in some kind of resonance, so that the forces involved mostly keep the planets in their orbits, and thus be stable for the long term. Co-orbits (as mentioned above) is such a resonance.
Oh, as about the effect of a perpendicularly orbiting planet. As I understand it, that would put a torque on the other planets and cause the whole system (or possibly just some of the system) to precess in a way they aren’t doing now.
Re: binary planet
Well having the mass of Earth effectively doubled could destabilize the system. I think it’s unlikely, but can’t be ruled out.
[quote]
And would you not have to change the solar-orbital period of Earth and its binary?
I don’t know why this would need to be done.
Actually, the perpendicular orbit might have one advantage. It would be unstable, but it seems likely that it would have a longer instability timescale than the in-plane unstable orbits. So it depends on how long you want your new planet to be good for.