My advice is don't act out yourself because she's acting out her feelings instead of verbalising them.
She's the teen and you are the adult. Don't take it personally, that'll immediately take you down the wrong path.
Have you considered that your DD might have issues that are only emerging now. Like that she has difficulty managing her feelings in general or is often overwhelmed. No one intends to be 'horrible' or 'vile'.
Do you ever sit down and try to talk to her like saying 'I see sometimes that you might be struggling with strong feelings. Is there anything I could do that might help with that?' Or 'I just wanted to let you know that I notice that and I'm trying to understand how difficult things are for you sometimes' or 'I see you slamming a door but that's not really giving me any information about what you are annoyed about'.
If she's pushing you then perhaps she's trying to 'move' you. She's trying to make you 'see' her. She doesn't feel like she is seen in the house. Maybe you and DH tend to only notice her when she's doing something wrong. So if that gets her noticed she'll keep doing things 'wrong'. Often happens in big families.
More positive attention might help, try catching her doing things right 3 times a day.
Even if she has been vile earlier and you addressed that, then don't hold the grudge. Immediately after you have addressed it with her, go back to your normal self asking her if she wants a cup of tea or has clothes for the wash...whatever.
Treat her as if she is the lovely child you know she is. Say goodnight no matter what. If you stop saying goodnight to her that's you acting out, but not actually saying anything to her. Because she'll feel even more abandoned otherwise.
Not saying goodnight won't teach her a lesson. Just like her slamming a door is not teaching you a lesson, but she's trying to express something she hasn't got words for.
All behaviour is communication. This is a good book
https://www.amazon.co.uk/How-Talk-Teens-will-Listen/dp/1853408573