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four incircles

Source: 2015 IMO Shortlist G7

July 7, 2016
IMO Shortlistgeometry

Problem Statement

Let ABCDABCD be a convex quadrilateral, and let PP, QQ, RR, and SS be points on the sides ABAB, BCBC, CDCD, and DADA, respectively. Let the line segment PRPR and QSQS meet at OO. Suppose that each of the quadrilaterals APOSAPOS, BQOPBQOP, CROQCROQ, and DSORDSOR has an incircle. Prove that the lines ACAC, PQPQ, and RSRS are either concurrent or parallel to each other.
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