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isosceles wanted, congurent triangles wanted, parallelograms, AD = BC

Source: 2008 Moldova JBMO TST 3.3

July 12, 2020
geometrycongruent trianglesparallelogramequal segmentsisosceles

Problem Statement

Let ABCDABCD be a convex quadrilateral with AD=BC,CD∦AB,AD∦BCAD = BC, CD \nparallel AB, AD \nparallel BC. Points MM and NN are the midpoints of the sides CDCD and ABAB, respectively. a) If EE and FF are points, such that MCBFMCBF and ADMEADME are parallelograms, prove that △BFN≡△AEN\vartriangle BF N \equiv \vartriangle AEN. b) Let P=MN∩BCP = MN \cap BC, Q=AD∩MNQ = AD \cap MN, R=AD∩BCR = AD \cap BC. Prove that the triangle PQRPQR is iscosceles.
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