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Prove that the circumcentres of the triangles are collinear

Source: IMO Shortlist 1997, Q9

August 10, 2008
geometryincentercircumcirclecollinearityIMO Shortlist

Problem Statement

Let A1A2A3 A_1A_2A_3 be a non-isosceles triangle with incenter I. I. Let Ci, C_i, i \equal{} 1, 2, 3, be the smaller circle through I I tangent to A_iA_{i\plus{}1} and A_iA_{i\plus{}2} (the addition of indices being mod 3). Let B_i, i \equal{} 1, 2, 3, be the second point of intersection of C_{i\plus{}1} and C_{i\plus{}2}. Prove that the circumcentres of the triangles A1B1I,A2B2I,A3B3I A_1 B_1I,A_2B_2I,A_3B_3I are collinear.
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