MEMO 2007 I3

Let $k$ be a circle and $k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $O_{1},O_{2},O_{3},O_{4}$ respectively, on $k$ . For $i = 1,2,3,4$ and $k_{5}= k_{1}$ the circles $k_{i}$ and $k_{i+1}$ meet at $A_{i}$ and $B_{i}$ such that $A_{i}$ lies on $k$ . The points $O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $k$ and are pairwise different.

Prove that $B_{1}B_{2}B_{3}B_{4}$ is a rectangle.