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Questionable parallelogram

Source: Sharygin CR 2022 P12

March 4, 2022
geometryparallelogram

Problem Statement

Let KK, LL, MM, NN be the midpoints of sides BCBC, CDCD, DADA, ABAB respectively of a convex quadrilateral ABCDABCD. The common points of segments AKAK, BLBL, CMCM, DNDN divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that ABCDABCD is a parallelogram?
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