We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schützenberger. Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {a^iba^j | 0 <= i+j <= d}. If X \subset B_d is a code, then |X| <= d+1. The second conjecture is due to Cerný and the author. Let A be an automaton with n states. If there exists a word of rank <= k in A, there exists such a word with length <= (n-k)^2.