Problem 670

17.29. Under sequential symmetries with respect to lines $l_{1}, \ldots, l_{n-1}$, which are perpendicular to the given lines and pass through the center of the circle, vertex $A_{1}$ of the desired polygon is transformed into vertex $A_{n}$. If $n$ is odd, the composition of these symmetries is a rotation by a known angle, so through point $M$ one needs to draw the chord $A_{1} A_{n}$ of known length. If $n$ is even, the considered composition is a symmetry with respect to some line, so from point $M$ one needs to drop a perpendicular to this line.