Question 157
An integer $n$ is called good if $n \ge 3$ and there are $n$ lattice points $P_1, P_2, \dots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $P_iP_j$ has a rational length, then there is $P_k$ such that both line segments $P_iP_k$ and $P_jP_k$ have irrational lengths; and if line segment $P_iP_j$ has an irrational length, then there is $P_k$ such that both line segments $P_iP_k$ and $P_jP_k$ have rational lengths.
(1) Determine the minimum good number.
(1) Determine the minimum good number.