Cutting cakes and kissing circles (Q6169272)

For \(n\geq 2\), suppose there are circles \(c_1, \dots, c_n\) on the plane such that \(c_i\cap c_{i+1}\) is a point \(S_i\) for \(1\leq i\leq n\), where \(c_{n+1} = c_1\). For \(P_1\in c_1\), let the straight line \(P_1S_1\) intersect \(c_2\) at \(P_2\). Inductively, let the straight line \(P_iS_i\) intersect \(c_{i+1}\) at \(P_{i+1}\). If \(n\) is even then \(P_{n+1}=P_1\) and if \(n\) is odd then \(P_1P_{n+1}\) is a diameter of \(c_1\). The author presents a nice sketch of a proof of this. If \(n\) is odd then changing the position of \(P_1\), one gets a different diameter of \(c_1\) and hence gets the center of \(c_1\). Using two tricks by \textit{J. Steiner} [Die geometrischen Constructionen, ausgeführt mittelst der geraden Linie und eines festen Kreises. Herausgegeben von A. J. von Oettingen. Frankfurt am Main: Harri Deutsch (1895; JFM 26.0545.01)], the author presents constructions to divide a circular disc with a marked center into \(m\) pieces of equal size where \(m=2, 3, 4\) and 6. The author presents this article in terms of cutting circular cakes into equal size pieces. More precisely, if there are odd numbers of circular cakes then using the above two constructions, one can cut any one of the cakes into \(m\) pieces of equal size where \(m=2, 3, 4\) and 6.