Question 19
Let $A_0, A_1, \dots, A_5$ be a circular labelling of six distinct points on a circle centred at the point $O$, and let $B_i$ be the midpoint of the segment $A_iA_{i+1}$, $i = 0, 1, \dots, 5$ (indices are reduced modulo $6$). Assume that no opposite sides of the hexagon $A_0A_1\cdots A_5$ are parallel. A line through the point $O$ meets again the circle $B_iOB_{i+3}$ at the point $C_i$, $i = 0, 1, 2$. Let $\ell_i$ be the tangent to the circle $B_iOB_{i+3}$ at the point $C_i$, $i = 0, 1, 2$. The lines $\ell_i$ and $\ell_j$ meet to produce the point $D_k$, where $\{i, j, k\} = \{0, 1, 2\}$. Show that the three circles $B_iOB_{i+3}$ and the circle $D_0D_1D_2$ share a point in the plane.