in fact we can prove a generalized version: Quote: If $k$ points on circle with parameter $k$, and The length of any arc between two points is greater than the number of points on the arc, then we can fit a $k$-regular gon in each arc Say two points on the circle, have $k$ $\frac{i}n$ and $l$ $\frac jm$ between them, then there distance $\frac{k+l}{m+n}<\max\{\frac in,\frac jm\},$ so use the above property and we are done. The proof can be found in attached file:

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