Most Helpful Guy
Don't tell him that you're jealous. Just ask him who she is, because you've noticed that she likes/loves his stuff on Facebook, and wait for his response and go from there. Then keep an eye on her Twitter and see if there are any changes to that.
Most Helpful Girl
Once I get through to the second half, it's obvious that something is very inappropriate about this situation. Since he's failed to bring it up to you, he's lost his right to privacy. I would check all of his messages to see what's actually going on and then confront him. If you confront him before doing this, he may just delete any and all evidence. It sounds like she is openly posting these things so you purposely see it. I would definitely get to the bottom of this