Why will she stare in my direction from far away, but when she catches me looking at her at a closer distance, she looks away before I even have a chance to smile? She is not shy around anyone, except possibly me.
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Well, if she likes you, she finds you intimidating. I get really self-conscious around my crush. People don't think I'm shy either, but quiet and self-conscious are two completely unrelated things. You see her being very talkative with her other friends and acquaintances because she has no reason to care what they think of her. The stakes are much lower. With you, on the other hand, she has to worry that she's not funny enough, not pretty enough, doesn't have enough interesting things to say, etc. She's afraid to look at you, because as they say, "The eyes are the window to the soul." Looking you in the eyes makes her very vulnerable and risks exposing her feelings, so she avoids extended eye contact.
The fact that you've noticed her change in behavior around you in particular is a good sign for you. Just start a casual conversation with her and see how she responds. You don't have to dramatically declare your love for her or anything. Just a simple, "Hey, I always bump into you here. How long have you been a personal trainer?" That way it could be perceived as friendly and nonthreatening, but could open the door for something else later.