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Concyclic if segments are diameters

Source: IMO Longlist 1989, Problem 6

September 18, 2008
geometryparallelogramgeometry unsolved

Problem Statement

The circles c1 c_1 and c2 c_2 are tangent at the point A. A. A straight line l l through A A intersects c1 c_1 and c2 c_2 at points C1 C_1 and C2 C_2 respectively. A circle c, c, which contains C1 C_1 and C2, C_2, meets c1 c_1 and c2 c_2 at points B1 B_1 and B2 B_2 respectively. Let ω \omega be the circle circumscribed around triangle AB1B2. AB_1B_2. The circle k k tangent to ω \omega at the point A A meets c1 c_1 and c2 c_2 at the points D1 D_1 and D2 D_2 respectively. Prove that (a) the points C1,C2,D1,D2 C_1,C_2,D_1,D_2 are concyclic or collinear, (b) the points B1,B2,D1,D2 B_1,B_2,D_1,D_2 are concyclic if and only if AC1 AC_1 and AC2 AC_2 are diameters of c1 c_1 and c2. c_2.
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