Why must isometry of regular polygon fix origin?

Question

Here is the question:

Suppose $\varphi\colon\Bbb R^2\to\Bbb R^2$ is an isometry and $\varphi(\pi_n)=\pi_n$, where $\pi_n$ is the regular $n$-gon with center at origin. Why must $\varphi$ fix the origin or the center?

Here, I don't want to use the fact that $\varphi$ is a permutation of vertices of $\pi_n$, because it is actually equivalent to the question I asked, and I don't know its proof either.

I think this question is really easy, because intuitively it is correct. However, I want to prove it rigorously and don't know how to proceed.

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Partial Answer:

I came up with a proof for $n=2q$ and am not sure if it is correct.

If $\varphi$ is an isometry and $S[A,B]$ is a line segment between $A\in\Bbb R^2$ and $B\in\Bbb R^2$, then $\varphi(S[A,B])=S[\varphi(A),\varphi(B)]$ and the length is also preserved. In particular, the midpoint of $S[A,B]$ is mapped to the midpoint of $S[\varphi(A),\varphi(B)]$. Take two vertices $v_i,v_j$ such that $O\in S[v_i,v_j]$ and consider four facts: (1) Every regular polygon has a circumscribed circle. (2) $S[v_i,v_j]$ is the diameter of the circumscribe circle. (3) Vertices of $\pi_n$ are on the circle. (4) $\varphi(\pi_n)=\pi_n$

Combining the four facts above, $\varphi(v_i)$ and $\varphi(v_j)$ must also be vertices of $\pi_n$, so that the midpoint of $S[\varphi(v_i),\varphi(v_j]$ is the center $O$. Therefore, the origin $O$ is preserved.

Answer 1

Approach 1: Circumcenter.

There is a unique point that is equidistant from all the vertices.

Approach 2: Incenter.

There is a unique point that is equidistant from all the sides.

Answer 2

A couple more approaches based on symmetry:

An isometry must take every axis of mirror symmetry to a similar axis, so must fix the unique point where the axes of symmetry intersect

And also

An isometry must take a centre of rotational symmetry to a similar centre. If there is only one such centre, it must be fixed.

These are not true for infinite configurations (wallpaper patterns, for example) where there can be an infinite number of axes/centres of symmetry which can be transformed into each other. It is the finiteness of the configuration which means that there is a single centre. If there is more than one, there must be an infinite number.