MathDB

4

Part of VMEO I 2004

Problems(1)

\lambda_A KA +\lambda_B KB + \lambda_C KC+ \lambda_D KD=0, in vectors

Source: 2004 VMEO I p4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__vietnam_mathematical

9/26/2021
In a quadrilateral ABCDABCD let EE be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ). https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png a) Prove that EK→=43EI→\overrightarrow{EK} =\frac43 \overrightarrow{EI}. b) Prove that λAKA→+λBKB→+λCKC→+λDKD→=0→\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0} , where λA=1+S(ADB)S(ABCD),λB=1+S(BCA)S(ABCD),λC=1+S(CDB)S(ABCD),λD=1+S(DAC)S(ABCD)\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)} , where SS is the area symbol.
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