Knight Zimmerman
06/12/2024 · Primary School
163 Soit \( A B C \) un triangle. Soit I le milieu du segment \( [A C] \) et \( G \) le point tel que: \[ \overrightarrow{G A}+2 \overrightarrow{G B}+\overrightarrow{G C}=\overrightarrow{0} \] 1. a. À l'aide de la relation de Chasles, montrer qu'on a l'éaglité \( 2 \overrightarrow{G B}+\overrightarrow{G C}=-3 \overrightarrow{A G}+2 \overrightarrow{A B}+\overrightarrow{A C} \).
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En utilisant la relation de Chasles, on exprime \( \overrightarrow{G B} \) et \( \overrightarrow{G C} \) en fonction de \( \overrightarrow{A} \), \( \overrightarrow{B} \) et \( \overrightarrow{C} \). En substituant ces expressions dans \( 2 \overrightarrow{G B} + \overrightarrow{G C} \) et en utilisant l'équation donnée pour \( \overrightarrow{G} \), on montre que \( 2 \overrightarrow{G B} + \overrightarrow{G C} = -3 \overrightarrow{A G} + 2 \overrightarrow{A B} + \overrightarrow{A C} \).
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