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trapezoid , 2PQ<=d wanted, AE=BF=CG=DH<AB/2 2018 Ecuador NMO (OMEC) 3.3

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September 18, 2021
geometrytrapezoid

Problem Statement

Let ABCDABCD be a convex quadrilateral with AB≤CDAB\le CD. Points E,FE ,F are chosen on segment ABAB and points G,HG ,H are chosen on the segment CDCD, are chosen such that AE=BF=CG=DH<AB2AE = BF = CG = DH <\frac{AB}{2}. Let P,QP, Q, and RR be the midpoints of EGEG, FHFH, and CDCD, respectively. It is known that PRPR is parallel to ADAD and QRQR is parallel to BCBC. a) Show that ABCDABCD is a trapezoid. b) Let dd be the difference of the lengths of the parallel sides. Show that 2PQ≤d2PQ\le d.
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