  In this paper we establish properties of independence for the continued
fraction expansions of two algebraic numbers. Roughly speaking, if the
continued fraction expansions of two irrational algebraic numbers have the same
long sub-word, then the two continued fraction expansions have the same tails.
If the two expansions have mirror symmetry long sub-words, then both the two
algebraic numbers are quadratic. Applying the above results, we prove a theorem
analogous to the Roth's theorem about approximation by algebraic numbers.
