I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read
An element $g$ of $D_n$ is a rigid motion taking the $n$-gon back to itself, and it must carry vertices to vertices (how are vertices unlike other points in terms of their distance relationships with all points on the polygon?) and $g$ must preserve adjacency of vertices.
I think I understand why $g$ must preserve adjacency of vertices: if $A$ and $B$ are vertices, then $g(A)$ and $g(B)$ are vertices, and the distance of $g(A)$ from $g(B)$ equals the distance of $A$ from $B$. Therefore, $g(A)$ and $g(B)$ are adjacent. Is this reasoning correct?
I think (but I'm not sure) that proving that $g$ preserves adjacency of vertices requires the fact that $g$ carries vertices to vertices. Why is that true? In particular, what is the answer to the author's question? (How are vertices unlike other points in terms of their distance relationships with all points on the polygon?) Does this question bear any relation to Lemma 2.1 in Conrad's notes?