How does one formally define ordering of $n$ points in a circle?

Question

Suppose $n$ is a positive integer greater than or equal to $4$. How does one formally define the statement, "Points $A_1,...,A_n$ are arranged in a circle in that order"? Note, I do not require that they be either clockwise or counterclockwise, in fact, clockwise and counterclockwise does not make sense in my question, because there is no coordinate system. I am merely requiring that they are in that order. I know it when I see it, but I am interested in how to define it mathematically in Euclidean geometry.

Answer 1

Here's one way to think about things that is completely coordinate-free and doesn't involve comparing lengths/angles/etc.:

Suppose I have a circle $C$ and distinct points $X,Y\in C$. We get two corresponding arcs in $C$ with endpoints $X$ and $Y$. (Explicitly, consider the equivalence relation $\approx$ on $C\setminus \{X,Y\}$ given by $U\approx V$ iff $\overline{UV}\cap \overline{XY}=\emptyset$.) Now given a sequence of points $\mathscr{A}=\langle A_1,...,A_n\rangle$ on a circle $C$, the sequence $\mathscr{A}$ is in order iff for each $1\le i<n$ one of the two arcs formed in $C$ by $A_i$ and $A_{i+1}$ contains no other point from $\mathscr{A}$.

Strictly speaking, (first-order) Euclidean geometry can't directly talk about finite sequences. So instead, the above paragraph should be thought of as a strategy for making sense of in-ordered-ness of length-$n$ sequenes for each fixed $n$.