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Bài 4.13 trang 105 SGK Toán 11 tập 1 - Cùng khám phá

Cho tứ diện ABCD. Gọi G, G’ lần lượt là trọng tâm của tam giác ABC, ACD. Chứng minh rằng đường thẳng GG’ song song với hai mặt phẳng (ABD) và (BCD).
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Đề bài

Cho tứ diện ABCD. Gọi G, G’ lần lượt là trọng tâm của tam giác ABC, ACD. Chứng minh rằng đường thẳng GG’ song song với hai mặt phẳng (ABD) và (BCD).

Phương pháp giải - Xem chi tiết

Chứng minh đường thẳng a không thuộc (P) song song với mặt phẳng (P): + Tìm đường thẳng b thuộc (P) sao cho a // b. + Suy ra a // (P).

Lời giải chi tiết

Gọi E là trung điểm AC\( \Rightarrow EG = \frac{1}{3}BE,EG' = \frac{1}{3}DE\)Xét tam giác EDB có \(\frac{{EG}}{{BE}} = \frac{{EG'}}{{DE}} = \frac{1}{3}\) nên GG’ // BDSuy ra GG’ // (BCD), GG’ // (BCD).

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