How do i compete with her?

I don't think it's worth your work for competing over this guy with this girl you've fallen out with- because the guy is obviously believing her over you, and there's a very good chance he may like her since he's now taking sides in this. I think it will be better for you to just drop the two of them- especially the girl you had a fall out with-not only after this fight will you not be friends again, but she's taking this guy away from you and on top of that is making him believe her and not you. And if the guy is going to be stupid enough to not think for himself then there's no point. There's not only better friends out there for you, but also better guys who will trust in what you say but who will also be able to think on their own instead of listening to others-especially liars.