(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.