Problem of the Week #3

(As usual, put solutions/suggestions in comments. A solution will be posted on June 12.)

Let k be an arbitrary circle. We then have four smaller circles k1, k2, k3, and k4, with all four circles having their centers O1, O2, O3, and O4 (respectively) on k. Suppose:
  • k1 and k2 intersect at two points A1 and B1, with A1 on circle k. 
  • k2 and k3 intersect at A2 and B2, with A2 on k. 
  • k3 and k4 intersect at A3 and B3, with A3 on k. 
  • k4 and k1 intersect at A4 and B4, with A4 on k.
 Finally, suppose that O1, A1, O2, A2, O3, A3, O4, and A4 lie in that order on k, and are all distinct. Prove that B1B2B3B4 is a rectangle.

(Source: Middle European Mathematical Olympiad 2007, Q4. But don't look! Solve it on your own.)

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