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OOH! I'm in the EXACT (save for age) situation. Unless you've talked to her a lot, it might be a bit creepy (she might think you're stalking her friend or something). My friend ended up being protective because she didn't want me hurting her friend--I have no idea where she got the idea that I would--and finally caved after 4 months of sucking up to the friend, probably not an ideal situation for you...
You could just say that you would just like to know some of her friends (multiple friends, to make your friend think it was just a coincidence you asked about a specific friend) over the summer and see how that goes. Alternatively, you could, instead of saying her personality intriguing, which your friend probably thinks is bs if you have to ask whether she's single or not, therefore, she would think you've never talked to her, say you were just messing around when you started asking questions and ask if while you're talking about her, get to know her friend and perhaps you and her friend would have more in common than your friend thought. You could also guilt trip your friend by saying, "Can't I just get to know some of your friends without being questioned about my intentions?" and say the asking her relationship status was just to weird your friend out and not meant to mean anything. Don't completely trust me though. I've been wrong MANY times before in relationships stuff. And I won't be able to reply back until after 3 PM...