[chrisser]

OK, is L3 stable or not?

Some terrible writing here. The author seems to imply that a planet on the far side of the sun couldn't exist because it's not in a Lagrange point, but then proceeds to show an image which diagrams a Lagrange point (L3) exactly there.

[Boogieman]

Yes, L3 is stable, but not for anything near to Earth’s mass. We could put an artificial satellite there, and it would stay opposite to Earth, but anything much larger than that would quickly drift into its own orbit.