Gandalf if it's any consolation my dd has hurled some fairly harsh insults at me too and they are always calibrated to hit the mark (ie directed at something I am particularly sensitive about). For a while, like Staywithmemyblood I was very low about it indeed and really took them to heart. I contemplated ADs too and haven't entirely dismissed the idea of taking them, even now.
Not swearing so much (well a tiny bit) but she said I was a crap mum, a shit mum, that all her friends' mums were more attractive and earned more (!), and most often that she hated me and wished I were dead ... oh and the best one .. I wish you hadn't survived your operation.
After that, I learned not to take her insults at face value because I know for sure that she doesn't hate me or want me dead but she may have done in that split second ifyswim. And she is allowed to dislike me intensely atm. After all, even the way I breathe is wrong! 
Sometimes I just think they feel to need to vent their general anger and frustration at life. Or they can't handle the way they think of themselves or their anxiety and so they hand it all over to us to "keep" for a bit. It's pretty wearing and demoralising but it serves a purpose sometimes I think and yes, Staywithmemyblood has the right idea with directing the anger elsewhere.
I also agree with Staywithmemyblood so much about the verbal attacks becoming a power struggle. That is what happened with dd and me and I should have learned to back out (not down!) much sooner. I honestly think she did it sometimes just because she was bored or wanted to provoke "something". Now, as soon as she starts, I walk out of the room. No hesitation or discussion. She knows that she can address whatever is bothering her later with me or dh when she can express it with civility.
We do the family meeting thing too which is more or less successful (the actions emanating from the meeting are more challenging to follow through).
And I must work on the "I feel that ...x and y" technique.
Sometimes I think extreme anger is a sign of frustration that they need more responsibility than they are currently getting or allowed to handle. So (in normal non-Covid times) I would suggest giving them a task that is slightly above their current ability and experience to manage and step back, and let them get on with it. This usually results in a more humble (in a good way) and respectful teen.
I wouldn't have dared to speak to my parents in the way dd speaks to me because I was too afraid of them. So perhaps they feel able to insult us - because they are emotionally closer to us - even though the behaviour is far from ideal of course and not to be encouraged. Also because they feel they have more agency within the family which is a good thing.
And just to reassure you, I can very much remember a situation where we had all had had a massive row with dd screaming at the top of her voice, and 45 mins later, following the arrival of guests, she was handing around canapes, joining in conversation, polite and engaged.
So comparisons are not necessarily helpful
Good luck 
Keep hanging on to the end of the rope. You are helping her so much by doing so, even though you are getting burns!
Complexneeds It isn't always easy NOT to have an objective-focused conversation when they have been non-communicative and moody for weeks and you have been storing up qs about the school trip payment, the time they need a lift to x or the amount of money they need for y. So don't be so hard on yourself!
Waves to Retrievers Staywithmemyblood and all PoTs!
Wishing you a calm and peaceful weekend everyone!