I saw a pretty girl and she looked at me. What should I do?

Anonymous
There is this girl in one of my college classes, she is very pretty. I've been in this class with her for about 3 weeks. And every time I enter the room, her and I make eye contact (although we have never talked and we don't sit by each other). It's a rather pleasant eye contact, we haven't smiled at each other but the contact feels warm and usually lasts a few seconds before one of us looks away.

After the second class I found out her name and I searched for her on bookface (to see if she is single, cause I'm too lazy [or most likely a coward] to get to know her, then ask her out on a date only to get rejected). Weird thing was I couldn't find her, even with her semi-unusual name.

So today I tried again but no luck. I thought that was weird since 95% of college students have an account. Even if they never use it, like me. Although, I found a woman who seemed could be her mother.

So naturally I googled her mother. Funny thing was within 10 min of googleing, I found out exactly where and when her parents got married and how old they were. I found out her address (registered under her father). So I then googled her address (I swear I'm not weird, I just really wanted to see her relationship status on the book of faces. I'm not gonna stalk her, I have the very best of intentions). I wish I could be a man and just talk to her.

Anywhosles, to make it short, I found out her real first name (all through google. It's so creepy what you can do with google), and then searched her on FB one more time and found her.

But I was bummed to find out she was in a relationship (her profile is set to private, so I can't see with whom. It could be one of those jokes where she is in a 'relationship' with her best friend or something).

If it was set as single, I would've made a move tomorrow. I've heard her speak, and her voice is so pleasant. It's not deep or very high or any bit annoying, it's very soothing. Also her eyes are really blue and nice to look at. When we make eye contact it's amazing. Why would a girl who is 'in a realtionship' make eye contact with me every time I enter the class? It's like teasing me.

It's hard in college, especially in a community college. Every semester it's different people, and one time or another I gotta pick myself up and ask a woman on a date. (I have never had a date at the age of twenty. Although I did have a girlfriend when I was 5, but doubt that counts).

Don't get me wrong, I am friendly with the pretty ladies. I flirt, they flirt back. But after we've know each other a bit, I don't make a move and I'm just a nice guy friend. The longer I wait to make a move the harder it seems to get.

I still really want to go talk to her, but knowing that we will most likely only end up friends (probably just till the end of the semester) turns me off.

So, should I go for it?. Or just continue searching? But I figure it's gonna be exactly the same every semester until the end of college. So why not make a move now.
I saw a pretty girl and she looked at me. What should I do?
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