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МатематикаРешебник по геометрии 11 класс. Атанасян ФГОС 621
621
\[\boxed{\mathbf{621.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]
\[Дано:\ \]
\[\ \mathrm{\Delta}ABC;\ \ \]
\[точки\ A_{1},B_{1}\ и\ C_{1} - середины\ \]
\[сторон\ BC,AC\ и\ AB;\]
\[O - произвольная\ точка\ \]
\[пространства.\]
\[Доказать:\ \ \]
\[\overrightarrow{OA_{1}} + \overrightarrow{OB_{1}} + \overrightarrow{OC_{1}} =\]
\[= \overrightarrow{\text{OA}} + \overrightarrow{\text{OB}} + \overrightarrow{\text{OC}}.\]
\[Доказательство.\]
\[1)\ \overrightarrow{OA_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{CA_{1}} = \overrightarrow{\text{OB}} + \overrightarrow{BA_{1}}:\ \]
\[2\overrightarrow{OA_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{CA_{1}} + \overrightarrow{\text{OB}} + \overrightarrow{BA_{1}};\]
\[\overrightarrow{CA_{1}} = - \overrightarrow{BA_{1}}.\]
\[Отсюда:\]
\[2\overrightarrow{OA_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{\text{OB}}.\]
\[2)\ \overrightarrow{OB_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{CB_{1}} = \overrightarrow{\text{OA}} + \overrightarrow{AB_{1}}:\]
\[2\overrightarrow{OB_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{CB_{1}} + \overrightarrow{\text{OA}} + \overrightarrow{AB_{1}};\]
\[\overrightarrow{CB_{1}} = - \overrightarrow{AB_{1}}.\]
\[Отсюда:\]
\[2\overrightarrow{OA_{1}} = \overrightarrow{\text{OC}} + \overrightarrow{\text{OA}}.\]
\[3)\ \overrightarrow{OC_{1}} = \overrightarrow{\text{OB}} + \overrightarrow{BC_{1}} = \overrightarrow{\text{OA}} + \overrightarrow{AC_{1}}:\]
\[2\overrightarrow{OC_{1}} = \overrightarrow{\text{OB}} + \overrightarrow{BC_{1}} + \overrightarrow{\text{OA}} + \overrightarrow{AC_{1}};\]
\[\overrightarrow{BC_{1}} = - \overrightarrow{AC_{1}}.\]
\[Отсюда:\]
\[2\overrightarrow{OC_{1}} = \overrightarrow{\text{OA}} + \overrightarrow{\text{OB}}.\]
\[4)\ Таким\ образом:\]
\[2\left( \overrightarrow{OA_{1}} + \overrightarrow{OB_{1}} + \overrightarrow{OC_{1}} \right) =\]
\[= 2\left( \overrightarrow{\text{OA}} + \overrightarrow{\text{OB}} + \overrightarrow{\text{OC}} \right).\]
\[Что\ и\ требовалось\ доказать.\]
