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Gọi N là trung điểm BC
\(\left|\overrightarrow{MA}+\overrightarrow{MC}+2\overrightarrow{MB}+2\overrightarrow{OC}\right|=\left|\overrightarrow{AB}-\overrightarrow{AD}\right|\)
\(\Leftrightarrow\left|2\overrightarrow{MO}+2\overrightarrow{MB}+2\overrightarrow{OC}\right|=\left|\overrightarrow{AB}-\overrightarrow{AD}\right|\)
\(\Leftrightarrow\left|2\overrightarrow{MC}+2\overrightarrow{MB}\right|=\left|\overrightarrow{AB}-\overrightarrow{AD}\right|\)
\(\Leftrightarrow4\left|\overrightarrow{MN}\right|=\left|\overrightarrow{BD}\right|\)
\(\Rightarrow\left|\overrightarrow{BD}\right|=4\left|\overrightarrow{MN}\right|=4\left|\overrightarrow{DN}+\overrightarrow{MD}\right|\ge4MD-4DN\)
\(\Rightarrow4MD\le BD+4DN\)
\(\Leftrightarrow MD\le\dfrac{BD+4DN}{4}=\dfrac{a\sqrt{2}+2a\sqrt{5}}{4}=\dfrac{2\sqrt{5}+\sqrt{2}}{4}a\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a)\(\overrightarrow{|AB}+\overrightarrow{OD|}=\left|\overrightarrow{AB}+\overrightarrow{BO}\right|=\left|\overrightarrow{AO}\right|=\frac{a\sqrt{2}}{2}\)
b)\(\left|\overrightarrow{MA}-\overrightarrow{MB}-\overrightarrow{MC}+\overrightarrow{MD}=\overrightarrow{BA}+\overrightarrow{CM}+\overrightarrow{MD}\right|=\left|\overrightarrow{BA}+\overrightarrow{CD}\right|=\left|\overrightarrow{BA}+\overrightarrow{BA}\right|=2a\)
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\(\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}+\overrightarrow{MD}=\overrightarrow{AB}+\overrightarrow{AD}\)
\(\Leftrightarrow\overrightarrow{MO}+\overrightarrow{OA}+\overrightarrow{MO}+\overrightarrow{OB}+\overrightarrow{MO}+\overrightarrow{OC}+\overrightarrow{MO}+\overrightarrow{OD}=\overrightarrow{AC}\)
\(\Leftrightarrow4\overrightarrow{MO}+\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=2\overrightarrow{AO}\)
\(\Leftrightarrow4\overrightarrow{MO}=2\overrightarrow{OA}\)
\(\Leftrightarrow\overrightarrow{MO}=\dfrac{1}{2}\overrightarrow{AO}\)
\(\Rightarrow M\) là trung điểm OA
![](https://rs.olm.vn/images/avt/0.png?1311)
1)\(VT=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=\overrightarrow{CO}+\overrightarrow{DO}+\overrightarrow{OC}+\overrightarrow{OC}=\overrightarrow{CO}+\overrightarrow{OC}+\overrightarrow{DO}+\overrightarrow{OD}=\overrightarrow{0}\)
2)\(VT=\overrightarrow{DA}-\overrightarrow{DB}+\overrightarrow{DC}=\overrightarrow{BA}+\overrightarrow{DC}=\overrightarrow{0}\)
3)\(VT=\overrightarrow{DO}+\overrightarrow{AO}=\overrightarrow{OB}+\overrightarrow{AO}=\overrightarrow{AB}\)
4)\(\overrightarrow{MA}+\overrightarrow{MC}=\overrightarrow{MB}+\overrightarrow{BA}+\overrightarrow{MD}+\overrightarrow{DC}=\overrightarrow{MB}+\overrightarrow{MD}\left(đpcm\right)=\overrightarrow{MO}+\overrightarrow{OB}+\overrightarrow{MO}+\overrightarrow{OD}=2\overrightarrow{MO}\left(đpcm\right)\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
b) \(VP=\overrightarrow{MC}-\overrightarrow{MD}=\overrightarrow{DC}=\overrightarrow{AB}=VP\left(đpcm\right)\)
c) \(\overrightarrow{BD}-\overrightarrow{BA}=\overrightarrow{OC}-\overrightarrow{OB}\\ \Leftrightarrow\overrightarrow{AD}=\overrightarrow{BC}\left(đúng\right)\\ \)
d) \(\overrightarrow{BC}-\overrightarrow{BD}+\overrightarrow{BA}=\overrightarrow{0}\\ \Rightarrow\overrightarrow{DC}+\overrightarrow{BA}=\overrightarrow{0}\\ \Leftrightarrow\overrightarrow{0}=\overrightarrow{0}\left(đúng\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
(*) mk mới hok dạng toán này trên mạng ; nên lm thử thôi nha bn
hình :
a) ta có : \(VT=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}+\overrightarrow{OE}+\overrightarrow{OF}\)
\(=\overrightarrow{OA}+\overrightarrow{EO}+\overrightarrow{OC}+\overrightarrow{AO}+\overrightarrow{OE}+\overrightarrow{CO}\)
\(=\left(\overrightarrow{AO}+\overrightarrow{OA}\right)+\left(\overrightarrow{CO}+\overrightarrow{OC}\right)+\left(\overrightarrow{EO}+\overrightarrow{OE}\right)\)
\(=\overrightarrow{AA}+\widehat{CC}+\overrightarrow{EE}=\overrightarrow{0}+\overrightarrow{0}+\overrightarrow{0}=\overrightarrow{0}=VP\left(đpcm\right)\)
b) ta có : \(VT=\overrightarrow{OA}+\overrightarrow{OC}+\overrightarrow{OE}=\overrightarrow{FO}+\overrightarrow{OE}-\overrightarrow{AO}\)
\(=\overrightarrow{FE}-\overrightarrow{FE}=\overrightarrow{EE}=\overrightarrow{0}=VP\left(đpcm\right)\)
c) ta có : \(VT=\overrightarrow{AB}+\overrightarrow{AO}+\overrightarrow{AF}=\overrightarrow{AB}+\overrightarrow{AF}+\overrightarrow{FE}\)
\(=\overrightarrow{AB}+\overrightarrow{AE}=\overrightarrow{AB}+\overrightarrow{BD}=\overrightarrow{AD}=VP\left(đpcm\right)\)
d) ta có : \(VT=\overrightarrow{MA}+\overrightarrow{MC}+\overrightarrow{ME}=\overrightarrow{MB}+\overrightarrow{BA}+\overrightarrow{MD}+\overrightarrow{DC}+\overrightarrow{MF}+\overrightarrow{FE}\)
\(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\left(\overrightarrow{BA}+\overrightarrow{DC}+\overrightarrow{FE}\right)\)
\(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\left(\overrightarrow{BA}+\overrightarrow{FE}+\overrightarrow{EO}\right)\) \(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\left(\overrightarrow{BA}+\overrightarrow{FO}\right)\) \(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\left(\overrightarrow{BA}-\overrightarrow{OF}\right)\) \(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\left(\overrightarrow{BA}-\overrightarrow{BA}\right)\) \(=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\overrightarrow{AA}=\left(\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}\right)+\overrightarrow{0}\) \(=\overrightarrow{MB}+\overrightarrow{MD}+\overrightarrow{MF}=VP\left(đpcm\right)\)