What do I do when I catch my crush staring at me, but she has a boyfriend?

At about January of 2015 I developed a crush on a girl, but she had a boyfriend (although I didn't know about it at the time). I let her catch me staring at her, and after a few weeks she started staring back, but she didn't want me to see her. Whenever I would catch her, she would quickly look away. At the last day of the school year, we were basically staring at each other the whole day, but we would look away quickly whenever it happened. Summer passed, and she still stared at me when school came back, but she didn't have a boyfriend anymore. But I am way too shy to even talk to her. After a few weeks she got a new boyfriend, but she kept on staring at me, and she still stares at me. Although she toned the staring down when she got the new boyfriend, but I still catch her doing it. She has told previous crushes that she likes them, so I'm not sure why she never told me. Maybe it is because she is popular, and I'm a semi-attractive nerd who doesn't even talk to her friend group. I should also probably mention that i have a reputation for being shy, and have never dated anyone. I'm not even sure why she goes out with the guy she's dating. He is odd. But he seems very attached to her, and I think he really likes her. But then again, all the guys like her. But, I am pretty sure she is playing him. Did she give me a chance at the beginning of the school year? Why does she like me, of all people? How can I at least become friends with her? Should I just wait this whole process out for a while to see if they break up, then try to start talking to her? I actually have a plan to get her phone number, but it kinda involves waiting 8 months, assuming she still likes me by that point. But it has been almost a year, and I have acted pretty weird when I get around her, and yet, she still stares, so...
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Also, she tried to become friends with one of my friends, so sometimes she sits with us. Is she trying to get to know me, or is she really just trying to become friends with him?
What do I do when I catch my crush staring at me, but she has a boyfriend?
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