Question

ABCD este paralelogram, {O} = AC intersectat cu BD. Sa se arate ca pentru orice punct M din planul paralelogramului are loc relatia: MA (vector) + MB (vector) + MC (vector) + MD (vector) = 4 * MO (vector).

Answer 1

$\overrightarrow{MA}=\overrightarrow{MO}+\overrightarrow{OA}\\\\\overrightarrow{MB}=\overrightarrow{MO}+\overrightarrow{OB}\\\\\overrightarrow{MC}=\overrightarrow{MO}+\overrightarrow{OC}\\\\\overrightarrow{MD}=\overrightarrow{MO}+\overrightarrow{OD}\\\\Stim\ ca\ intr-un\ paralelogram\ intersectia\ diagonalelor\\\ este\ mijloc\ pentru ambele\ segmente\ atunci\ \overrightarrow{OA}=\overrightarrow{BO}=-\overrightarrow{OB}\\\\\overrightarrow{OC}=\overrightarrow{DO}=-\overrightarrow{OD}$

$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}+\overrightarrow{MD}=4*\overrightarrow{MO}+\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}\\\\=4*\overrightarrow{MO}-\overrightarrow{OB}+\overrightarrow{OB}-\overrightarrow{OD}+\overrightarrow{OD}\\\\=4*\overrightarrow{MO}$