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Câu 3 trang 91 SGK Hình học 11 Nâng cao

Cho hình lăng trụ tam giác ABC.A’B’C’. Gọi G và G’ lần lượt là trọng tâm của tam giác ABC và A’B’C’, I là giao điểm của hai đường thẳng AB’ và A’B. Chứng minh rằng các đường thẳng GI và CG’ song song với nhau.
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Đề bài

Cho hình lăng trụ tam giác ABC.A’B’C’. Gọi G và G’ lần lượt là trọng tâm của tam giác ABC và A’B’C’, I là giao điểm của hai đường thẳng AB’ và A’B. Chứng minh rằng các đường thẳng GI và CG’ song song với nhau.

Lời giải chi tiết

 

Đặt \(\overrightarrow {AA'}  = \overrightarrow a ,\overrightarrow {AB}  = \overrightarrow b ,\overrightarrow {AC}  = \overrightarrow c \)Thì \(\overrightarrow {AG}  = {1 \over 3}\left( {\overrightarrow b  + \overrightarrow c } \right),\overrightarrow {AI}  = {1 \over 2}\left( {\overrightarrow a  + \overrightarrow b } \right)\)Do đó, \(\overrightarrow {GI}  = \overrightarrow {AI}  - \overrightarrow {AG}  = {{3\overrightarrow a  + \overrightarrow b  - 2\overrightarrow c } \over 6}\)Mặt khác : \(\overrightarrow {AG'}  = {1 \over 3}\left( {\overrightarrow {AA'}  + \overrightarrow {AB'}  + \overrightarrow {AC'} } \right) = \overrightarrow a  + {1 \over 3}\left( {\overrightarrow b  + \overrightarrow c } \right)\)\( \Rightarrow \overrightarrow {CG'}  = \overrightarrow {AG'}  - \overrightarrow {AC}  = \overrightarrow a  + {1 \over 3}\left( {\overrightarrow b  + \overrightarrow c } \right) - \overrightarrow c  \)               \(= {{3\overrightarrow a  + \overrightarrow b  - 2\overrightarrow c } \over 3}\)
Vậy \(\overrightarrow {CG'}  = 2\overrightarrow {GI} .\) Ngoài ra, điểm G không thuộc đường thẳng CG’ nên GI và CG’ là hai đường thẳng song song.

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