Trichotillomania (Hair Pulling) Support Group
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I am quite a fan of Piaf also! I saw the movie and cannot wait to get the DVD. I, too, noticed that she did not have any eyebrows, though I didn't think about Trich as a possible reason for the absence of them until you mentioned it.
I looked around on google for an answer as to if she had Trich of not, but wasn't able to find anything. Let me know if you're able to come up with an answer.
http://trichotillomaniacs.blogspot.com/2016/01/8-celebrities-with-trichotillomania.html
Here's the link!