Hmm. We're in a situation where, if you could guess what I'm after, the verification is easy, but getting the correct guess is not so easy.

I explained what I meant by symmetry in the sentence or two after the one you quoted. However, if it's not helpful, try this instead: solving the equation in post # 4 for b, and for c if you feel like it. The algebra is going to be much the same as what you did for a. You should get one more unique possible way to get the equation to work. What do you get?