Prove that for every positive integer $n$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $n$ points in $S$ at unit distance from $A$.
I tried inducting on $n$. For $n=1$, we can simply take a line segment of length $1$. For $n=2$ we can take an equilateral triangle of side length $1$, but the problems starts from $n=3$, can somebody tell me how to get such a structure? This might help me inducting.