Is it normal that i am in "love" with a girl that i broke up with 9 months ago?

So, I'm not going to lie, I think I still "Love" with a girl that I broke up with nine months prior to me posting this. So we went out before, but we broke up. Then we went out again, after pretending to like each other so this creep would leave her alone at a New Year's party 2 years ago. From there we started talking until finally, one day I asked her out in March. Fast Forward 3 months, and I broke up with her, because I thought she was unhappy with me, when it turns out she wasn't. I broke her heart. Fast Forward 5 months, she tells me she likes me again, yet she then starts talking to some other guy. She knows they aren't going to be serious, but she goes for it anyway. I, the fool, still like her. Then later, he breaks her heart. and I tell her I like her again. I ask her to prom. She says yes. She even likes me. Then here we are nowish. We can't have a serious conversation, we are always fighting/flirting, but it always ends up with us getting frustrated. She says I blow her off and such. So she starts talking to some other guy, and once again, I still like her. We even break off our prom ordeal. Now we are talking again, and flirting still but I don't know. I honestly don't see why I like her still, but I do. She's great, and in a weird way, perfect. Helk, she's even White and I'm Hispanic, and it still worked. I dunno. It seems as though, she's somehow managed a way to keep herself in my mind. So I ask, is this normal?
P.S: The ironically funny thing is, that I am honestly a skeptic of love.