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incenters of XYZ and ABC are collinear - ISL 1986

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August 31, 2010
geometryincentercircumcirclecollinearityIMO Shortlist

Problem Statement

The circle inscribed in a triangle ABCABC touches the sides BC,CA,ABBC,CA,AB in D,E,FD,E, F, respectively, and X,Y,ZX, Y,Z are the midpoints of EF,FD,DEEF, FD,DE, respectively. Prove that the centers of the inscribed circle and of the circles around XYZXYZ and ABCABC are collinear.
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