Should I break up with my boyfriend because he is best friends with a girl?

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Okay so everytime I try to bring it up to her to talk about it she's like "OMG Alyssa were just friends stop getting mad for stupid shit!" Like idk what to do I mean I trust him a lot but I don't trust her and I'm not the only one who thinks she likes him but whenever I ask she says no is she lying to me? Like idk what to do and sometimes she's like "fine I won't be his friend anymore.." I mean I do get mad when they are not friends cause they seem sad to me so it makes me feel like I'm the bad guy but at the same time I hate them being friends it's like so hard and she has him in her bio and it's "Eduardo😏" and she sometimes posts their convo on Instagram is she doing that on purpose ? Like Idk what to think because at first I don't want them to be friends and I told her that and she like fine I won't but then I get mad because I feel like I'm the bad guy and they always seem sad when they are not friends like IDK WTF do please help me