jmc

To visit all four points, we notice that we must travel along at least three different segments. The sum of the shortest three segments is $3+4+5=12$, but we quickly notice that it is impossible to start at one point and visit the other three points by traveling on a path with length $12$ ($DB$, $BC$, and $CD$ do not let us visit point $A$, and it is not possible to travel on $AD$, $CD$, and $BD$ in a continuous path). We now look for a path with length $13$ and notice that traveling from point $A$ to $D$ to $B$ to $C$ works. Alternatively, $B$ to $D$ to $C$ to $A$ also works. Both paths have length $\boxed{13}$.