I really feel like it can be hard to find advice on social circle game. Since she is a friend of some friends of mine, or i guess loosely my own friend, i figured the opening phase is done and this is most appropriately a mid-phase question.

Now, my situation is not too complex so hopefully someone here can relate and guide me in what my next step should be. To shortly explain my situation there is this girl I am interested in a lot. I know her because we share a few of the same friends so we occasionally wind up at the same parties or other places with these mutual friends. She knows me well enough to approach me and hug me at these parties, but no well enough that we would coordinate plans specifically with each other. (don't even have her number yet or anything.) Our conversations are not usually too long, so I am asking for help on what to do after we greet each other? How can I transition to building attraction in this situation? I feel like most of the things you would do in a cold approach come off awkward or like I'm trying too hard with someone that you already know. I'm worried that it would be easy to wind up in the friend zone in this situation unless I act fast because shes seen me enough times to at least feel comfortable around me and maybe even write me off. Would you say this is true? or should I actually take a longer approach with someone in my extended social circle? I'm going to school soon so maybe wait until christmas break?

Now I'll add the complex element but feel free to ignore this and just guide me in my Mid-Game with this woman.
Unfortunately, It seems that she is in love with one of my best friends. Shes been close with him way longer than I have so I understand her feelings. He's been kind of a jerk to her though and whenever I see her she tells me how much she hates him, but I can tell she still loves him because she'll just wind up asking him to take her to ice cream or something the next day. The thing is he supports me trying to have her. He's not attracted to her and thinks of her as nothing more than a close friend. Does this change anything? Is there a way to shift her focus from him onto me? Or should I just ignore it and go for her assuming she has no feeling for anyone at the moment?

Any advice would be much appreciated though so I know what I should consider doing when I see her in the following weeks.

Thanks