@UnflatteringComment I'm sure, deep down, your DSD understands and appreciates this, but that doesn't mean your weightloss hasn't been hard for her. It's hard for me when people around me lose weight, and I'm an adult.
It's been a long time since I was 16, but I have a vague recollection of wholly believing everyone was against me, no one understood me, no one liked me, and that woman at Tesco who sighed when I walked past definitely hated me for no reason, and it wasn't fair. I'm pretty sure that wasn't my illness; it was just part and parcel of being 16.
As an adult, I am happy for my friends when they lose weight, if it's making them happy and healthier, I truly want them to succeed, but there is still that small, ED part of me that finds it hard.
They're doing the two things I love more than anything else in the world that I am not allowed to do - controlling what they eat and losing weight. It does make me envious. Logically, I know that restricting what I eat is unhealthy for me in a way that it is not for other people, but that doesn't mean I don't miss it or wish I could manage to do it the way other people do.
And there is still that small voice that tells me they're doing it better than I do, and everyone can see it, and that is a reflection on me. It means I am not good enough, or valuable enough, and that I don't deserve to be happy or healthy. Again, logically, I know none of this is true, but it's not always easy to believe.
I don't think I've ever said anything disparaging out loud about other people's bodies, but I have thought things to myself. It's not because I believe it, or because I am judging them, it's my way of making myself feel better about myself, if that makes sense? I might not be as good at losing weight as they are or as valuable as they are as a person, but at least my arse looks better than theirs.
I don't know why DSD said what she did. I don't know if it was her illness or just because she is a teenager and teenagers are mean. I don't think she went running to her dad to say that you commented on her weight to get you in trouble or to save face; I honestly think she said that because she believes it to be true.
You had every right to be upset at what they said, and to tell them that, and they had no right to comment on your body, but that doesn't change the fact that your reaction was out of proportion to the whole context and potentially very damaging to your DSD's health and her relationship with you.
I don't know your DSD, and everyone deals with recovery in their own way. I hate people bringing up my illness when I'm struggling, but I know others who like to talk about it and feel it helps them.
A good starting point, if you want to try to fix things, would be to send her a text to apologise (explain that she hurt your feelings if you want to), tell her you didn't mean it, you were hurt and angry because of what you heard, and that you love her. After that, the ball's in her court. She might respond, she might not, but it will help her.