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MMO 331 Moscow MO 1956 A_1B_1/B_1A_2 . A_2B_2/B_2A_3... A_nB_n/B_nA_1=1

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August 19, 2019
broken line3D geometrygeometryProduct

Problem Statement

Given a closed broken line A1A2A3...AnA_1A_2A_3...A_n in space and a plane intersecting all its segments, A1A2A_1A_2 at B1,A2A3B_1, A_2A_3 at B2B_2 ,...... , AnA1A_nA_1 at BnB_n, prove that A1B1B1A2⋅A2B2B2A3⋅A3B3B3A4⋅...⋅AnBnBnA1=1\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1.
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