Phase equilibrium in mixtures
The solution is a direct application of the phase rule
$$F=C+2-P$$
with the number of components C=3 and F the number of degrees of freedom (the number of intensive variables that can be varied while phases continue to coexist). The phase rule can be derived by considering all the conditions of balance (chemical, pressure, temperature) that must be satisfied by the phases in mutual equilibrium.
Solving for P (the number of phases).
$$P=5-F$$
The maximum number of coexisting phases is observed when there are no degrees of freedom left (a coexistence point), that is when
$$P=5$$