I found the following problem interesting:
In a three villages $A$, $B$ and $C$ there are $a,b$ and $c$ pupils respectively. Where should one build the school such that the total length of pupils going to the school is as small as possible?
How one could solve it as there are no coordinates of villages given?
This is Fermat point. See the link below
https://en.wikipedia.org/wiki/Fermat_point
For weighted sum, it is usually called weighted Fermat point. Make a search on google.