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6 circles, other concentric other tangent, 8 intersections lie in 2 circles

Source: Sharygin 2007 Final 10.6

April 30, 2019
circlesgeometrycombinatorial geometry

Problem Statement

Given are two concentric circles Ω\Omega and ω\omega. Each of the circles b1b_1 and b2b_2 is externally tangent to ω\omega and internally tangent to Ω\Omega, and ω\omega each of the circles c1c_1 and c2c_2 is internally tangent to both Ω\Omega and ω\omega. Mark each point where one of the circles b1,b2b_1, b_2 intersects one of the circles c1c_1 and c2c_2. Prove that there exist two circles distinct from b1,b2,c1,c2b_1, b_2, c_1, c_2 which contain all 88 marked points. (Some of these new circles may appear to be lines.)
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