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Relationships between triangles and parallelograms

Source: Poland Math Olympiad 1993 Round 1 #7

June 3, 2023
geometrysimilar trianglesparallelogram

Problem Statement

Given convex quadrilateral ABCDABCD. We construct the similar triangles APB,BQC,CRD,DSAAPB, BQC, CRD, DSA outside ABCDABCD so that ∠PAB=∠QBC=∠RCD=∠SDA,∠PBA=∠QCB=∠RDC=∠SAD\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD. Prove that if PQRSPQRS is a parallelogram, so is ABCDABCD.
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