Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this method, but it doesn't look obvious to me... and I haven't been able to find one on the internet.

The algorithm: draw a circle of center $O$ and diameter $AB$, which is partitioned in $N$ parts $A=X_0$ and $B=X_N$ (using a ruler or any other method). Draw the equilateral triangles $ABC$ and $ABD$. Then the book claims that the intersections $X'_i$ of the circle and lines $CX_{2i}$ outside of segment $CX_{2i}$, $i\leq N/2$ and same with $DX_{N-2i}$, define an equipartition of the circle circumference.

I have tried to prove it starting with the dot product of $OX'_i$ and $OX'_{i+1}$ and trying to show it's not dependent on $i$, but have not succeeded.

Is there a classical proof? Ideas?