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Romania 2

Source: IMO LongList 1959-1966 Problem 17

September 1, 2004
geometryparallelogramvectorLocus problemsLocusIMO LonglistIMO Shortlist

Problem Statement

Let ABCDABCD and A′B′C′D′A^{\prime }B^{\prime}C^{\prime }D^{\prime } be two arbitrary parallelograms in the space, and let M,M, N,N, P,P, QQ be points dividing the segments AA′,AA^{\prime }, BB′,BB^{\prime }, CC′,CC^{\prime }, DD′DD^{\prime } in equal ratios. a.) Prove that the quadrilateral MNPQMNPQ is a parallelogram. b.) What is the locus of the center of the parallelogram MNPQ,MNPQ, when the point MM moves on the segment AA′AA^{\prime } ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.
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