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parallelogram criterion with trisection of a diagonal

Source: Mexican Mathematical Olympiad 1996 OMM P1

July 28, 2018
geometryparallelogram

Problem Statement

Let PP and QQ be the points on the diagonal BDBD of a quadrilateral ABCDABCD such that BP=PQ=QDBP = PQ = QD. Let APAP and BCBC meet at EE, and let AQAQ meet DCDC at FF. (a) Prove that if ABCDABCD is a parallelogram, then EE and FF are the midpoints of the corresponding sides. (b) Prove the converse of (a).
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