  Probabilistically creating n perfect clones from m copies for one of N priori
known quantum states with minimum failure probability is a long-standing
problem. We provide a rigorous proof for the geometric approach to this
probabilistic quantum cloning problem when N = 2. Then, we give the general
geometric form of the sufficient and necessary condition of probabilistic
cloning for N known quantum states. By this general geometric approach, we
realize the optimal probabilistic quantum cloning of N known quantum states
with priori probabilities. The results are also applicable to the
identification of those N quantum states.
