KöMaL 2023 A868

A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between $100/101$ and $101/100$, or at least 100 or at most $1/100$. Is it true that for any distinct points $A_1, A_2,\ldots, A_n$ in the plane it is always possible to find distinct points $A'_1, A'_2,\ldots,A'_n$ that form a disharmonic set of points, and moreover $A_i$, $A_j$ and $A_k$ are collinear in this order if and only if $A'_i$, $A'_j$ and $A'_k$ are collinear in this order (for all distinct $1\le i,j,k\le n$)?