7.73 As shown in the map of an industrial area, a highway (thick line) passes through this area, connecting seven factories $A_{1}, A_{2}, \cdots, A_{7}$, which are distributed on both sides of the highway and connected to it by some small roads (thin lines). Now, a long-distance bus station needs to be set up on the highway, with the goal of minimizing the total distance from the station to each factory (along the highway and small roads).
(1) Where should this station be set up for the best result?
(2) Prove your conclusion.
(3) If another factory is established at $P$ and a small road is built along the dashed line in the figure, where should the station be set up then?
[Solution 1] Let $B, C, D, E, F$ be the intersections where the small roads connect to the highway.
(1) Setting the station at point $D$ is the best.
(2) If the station is set at some point $S$ north of $D$ on the highway. Let $u_{1}, u_{2}, \cdots, u_{7}$ represent the distances from $S$ to each factory,
$$
w=u_{1}+u_{2}+\cdots+u_{7}.
$$

When $S$ moves a distance $d$ towards $C$, $u_{1}, u_{2}$ each decrease by $d$, so $w$ increases by $5d - 2d = 3d$.

When $S$ moves a distance $d'$ from $C$ towards $B$, $u_{1}$ decreases by $d'$, and $u_{2}, \cdots, u_{7}$ each increase by $d'$, so $w$ increases by $6d' - d' = 5d'$.

If $S$ moves a distance $d''$ north from $B$, $w$ increases by $7d''$, which means that $S$ at any point north of $D$ is not as good as at $D$. Similarly, it can be proven that $S$ at any point south of $D$ is not as good as at $D$.
(3) Setting the station at $D$, $E$, or any point between $D$ and $E$ is acceptable.
[Solution 2] First, the station should not be north of $B$, otherwise, people from each factory would have to walk the additional distance north of $B$. Similarly, the station should not be south of $F$, so the station should be set between $B$ and $F$.

Second, regardless of where the station is set between $B$ and $F$, the total distance walked on the highway by people from $A_{1}$ and $A_{7}$ is a constant, equal to the distance from $B$ to $F$. Since $A_{1}$ and $A_{7}$ together must walk this distance, we can disregard $A_{1}$ and $A_{7}$, so the station should be set between $C$ and $F$.

Third, similarly, regardless of where the station is set between $C$ and $F$, $A_{2}$ and $A_{6}$ together must walk the distance from $C$ to $F$, so we can also disregard $A_{2}$ and $A_{6}$, and only consider the distance walked by people from $A_{3}, A_{4}, A_{5}$ between $D$ and $E$.

Fourth, as long as the station is south of $D$, for example, a distance $d$ from $D$ (shorter than $DE$), $A_{3}, A_{4}, A_{5}$ must walk a distance of $DE + d$. Therefore, when $d = 0$, i.e., the station is set at $D$, it is the best.
For the addition of factory $P$, the conclusion is the same as in Solution 1.