KöMaL 2011 A547

A tetrahedron $OA_1 A_2 A_3$ is given. For $i$=1,2,3 let $B_i$ be a point in the interior of the edge $OA_i$ and let $C_i$ be a point on the ray $OA_i$, beyond $A_i$. Suppose that the polyhedron bounded by the six planes $OA_{i+1} A_{i+2}$ and $B_i A_{i+1} A_{i+2}$ ($i$=1,2,3) circumscribes a sphere, and the polyhedron bounded by the planes $B_i A_{i+1} A_{i+2}$ and $C_i A_{i+1} A_{i+2}$ also circumscribes a sphere. Prove that the polyhedron bounded by the planes $OA_{i+1} A_{i+2}$ and $C_i A_{i+1} A_{i+2}$ also circumscribes a sphere.