Thank you everyone. I have given her the sentence "yeah, ok... Anyone seen any good films lately?" to repeat and repeat and repeat, without edge, so it just shows she has the measure of her mother, without being cheeky enough to warrant being kicked out of the house again. She finds it hard to ignore it when her mother tries to pick a fight as this almost 50 year old woman apparently likes to get up in people's faces and scream abuse at them.
DSD mustn't get drawn into a discussion of validation or otherwise with her mother - she is perfectly healthy, eats well, exercises at school and is lovely and slim. She is 14, BTW, and a really lovely, thoughtful, sensible, funny, smart and kind child who I like very much. She is sweet to DS, who is 3.
We are trying to let her see, without directly telling her, that her mother is not able to have a healthy, respectful relationship with her, but when you're 14, it's probably impossible to believe that your mum doesn't actually love you unconditionally.
There is too much background to go into here - the jealousy comment (of everyone in her life, no matter how tangential, not just DSD) is spot on and very insightful. Suffice to say, the reason she has been NC with her mother is because the therapist said they should remain apart until her mother sought some therapy of her own, which hasn't happened.
So I am standing by the phone in case we need to run to her rescue - have lost count of the number of times her mother has thrown her out now, screaming "this is not your home, you are not welcome here"... Fingers crossed today will be different as DP is at work and I can't imagine what would happen if I pulled up with DS in the car.
Poor DSD, thank you all so much for your replies.