How can I prove that it is impossible to find a cheater in shamir secret sharing? Three participants Quadratic polynomial, one cheater. The Cheater discloses a false (x,y) pair I need to prove that any one of the truth-telling participants can't tell which of the others is the liar. I do realize that any pair of participants have a linear subspace of possible solutions since they don't know the secret they can't tell if it is a member of their subspace, but I don't think it is a proof just an intuition as to why this claim is true.
You could use the fact that having two of three pairs does not reveal any information about the secret (polynomial). Also see the Wikipedia article on Shamir's secret sharing: it provides you with a nice graphical answer to your question.