Let
be a circle and
four smaller circles with their centres
respectively, on
. For
and
the circles
and
meet at
and
such that
lies on
. The points
lie in that order on
and are pairwise different.
Prove that
is a rectangle.
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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.
Školjka