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2012 ToT Fall Senior A p4 A_1BC_1I is cyclic then <AKC is obtuse

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March 22, 2020
Cyclicobtusemidpointincentergeometry

Problem Statement

In a triangle ABCABC two points, C1C_1 and A1A_1 are marked on the sides ABAB and BCBC respectively (the points do not coincide with the vertices). Let KK be the midpoint of A1C1A_1C_1 and II be the incentre of the triangle ABCABC. Given that the quadrilateral A1BC1IA_1BC_1I is cyclic, prove that the angle AKCAKC is obtuse.
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