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a/ \(\left\{{}\begin{matrix}\overrightarrow{AB}+\overrightarrow{AC}=2\overrightarrow{AE}\\\overrightarrow{AM}+\overrightarrow{AN}=2\overrightarrow{AE}\end{matrix}\right.\) \(\Rightarrow\overrightarrow{AB}+\overrightarrow{AC}=\overrightarrow{AM}+\overrightarrow{AN}\)
b/ \(2\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}=2\overrightarrow{IA}+2\overrightarrow{IE}=2\left(\overrightarrow{IA}+\overrightarrow{IE}\right)=2\overrightarrow{0}=\overrightarrow{0}\)
c/ \(2\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=2\left(\overrightarrow{OI}+\overrightarrow{IA}\right)+\overrightarrow{OI}+\overrightarrow{IB}+\overrightarrow{OI}+\overrightarrow{IC}\)
\(=\left(2\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}\right)+4\overrightarrow{OI}=\overrightarrow{0}+4\overrightarrow{OI}=4\overrightarrow{OI}\)
Ta có: \(\overrightarrow{IA}-2\cdot\overrightarrow{IB}+4\cdot\overrightarrow{IC}=\overrightarrow{0}\)
=>\(\overrightarrow{IA}-2\left(\overrightarrow{IA}+\overrightarrow{AB}\right)+4\left(\overrightarrow{IA}+\overrightarrow{AC}\right)=\overrightarrow{0}\)
=>\(3\cdot\overrightarrow{IA}-2\cdot\overrightarrow{AB}+4\cdot\overrightarrow{AC}=\overrightarrow{0}\)
=>\(3\cdot\overrightarrow{IA}=2\cdot\overrightarrow{AB}-4\cdot\overrightarrow{AC}\)
=>\(\overrightarrow{IA}=\frac23\cdot\overrightarrow{AB}-\frac43\cdot\overrightarrow{AC}\)
\(P=\overrightarrow{IA}\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(=\left(\frac23\cdot\overrightarrow{AB}-\frac43\cdot\overrightarrow{AC}\right)\left(\overrightarrow{AB}+\overrightarrow{AC}\right)=\frac23\cdot\left(\overrightarrow{AB}\right)^2-\frac23\cdot\overrightarrow{AB}\cdot\overrightarrow{AC}-\frac43\cdot\left(\overrightarrow{AC}\right)^2\)
\(=\frac23\cdot AB^2-\frac23\cdot AB\cdot AC\cdot cosBAC-\frac43\cdot AC^2\)
\(=\frac23\cdot AB^2-\frac23\cdot AB^2\cdot cos60-\frac43\cdot AB^2=-\frac23\cdot AB^2-\frac23\cdot AB^2\cdot\frac12\)
\(=-AB^2=-a^2\)
Đặt \(\overrightarrow{BF}=x.\overrightarrow{BC}\)
D là trung điểm AC \(\Rightarrow\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{BA}+\dfrac{1}{2}\overrightarrow{BC}=-\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{BC}\)
DE=3BE \(\Rightarrow\overrightarrow{BE}=\dfrac{1}{4}\overrightarrow{BD}=-\dfrac{1}{8}\overrightarrow{AB}+\dfrac{1}{8}\overrightarrow{BC}\)
Ta có:
\(\overrightarrow{AE}=\overrightarrow{AB}+\overrightarrow{BE}=\overrightarrow{AB}-\dfrac{1}{8}\overrightarrow{AB}+\dfrac{1}{8}\overrightarrow{BC}=\dfrac{7}{8}\overrightarrow{AB}+\dfrac{1}{8}\overrightarrow{BC}=\dfrac{7}{8}\left(\overrightarrow{AB}+\dfrac{1}{7}\overrightarrow{BC}\right)\)
\(\overrightarrow{AF}=\overrightarrow{AB}+\overrightarrow{BF}=\overrightarrow{AB}+x.\overrightarrow{BC}\)
Mà A, E, F thẳng hàng
\(\Rightarrow x=\dfrac{1}{7}\Rightarrow BF=\dfrac{1}{7}BC\Rightarrow\dfrac{BF}{FC}=\dfrac{1}{6}\)
1) Có \(2\overrightarrow{EF}=\overrightarrow{ED}+\overrightarrow{EC}\)
Lại có : \(\left\{{}\begin{matrix}\overrightarrow{AD}=\overrightarrow{AE}+\overrightarrow{ED}\\\overrightarrow{BC}=\overrightarrow{BE}+\overrightarrow{EC}\end{matrix}\right.\rightarrow\overrightarrow{AD}+\overrightarrow{BC}=\left(\overrightarrow{AE}+\overrightarrow{BE}\right)+\overrightarrow{ED}+\overrightarrow{EC}=\overrightarrow{0}+\overrightarrow{ED}+\overrightarrow{EC}=\overrightarrow{ED}+\overrightarrow{EC}\) Do đó : \(2\overrightarrow{EF}=\overrightarrow{AD}+\overrightarrow{BC}\left(=\overrightarrow{ED}+\overrightarrow{EC}\right)\)
2) Có : \(\left\{{}\begin{matrix}\overrightarrow{OA}+\overrightarrow{OB}=2\overrightarrow{OE}\left(1\right)\\\overrightarrow{OC}+\overrightarrow{OD}=2\overrightarrow{OF}=-2\overrightarrow{OE}\left(2\right)\end{matrix}\right.\)
(1) + (2) => \(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=2\overrightarrow{OE}+2\overrightarrow{OF}=2\overrightarrow{OE}-2\overrightarrow{OE}=\overrightarrow{0}\)
3) \(\left(\overrightarrow{AB}+\overrightarrow{AD}\right)+\overrightarrow{AC}=2\overrightarrow{AC}=4\overrightarrow{AO}\)
4) Ta có : \(\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}+\overrightarrow{MD}=\left(\overrightarrow{MO}+\overrightarrow{OA}\right)+\left(\overrightarrow{MO}+\overrightarrow{OB}\right)+\left(\overrightarrow{MO}+\overrightarrow{OC}\right)+\left(\overrightarrow{MO}+\overrightarrow{OD}\right)=4\overrightarrow{MO}+\left(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}\right)=4\overrightarrow{MO}+\overrightarrow{0}=4\overrightarrow{MO}\)