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Giải bài 3 trang 112 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Cho tứ diện ABCD. Gọi E, F, G lần lượt là các điểm thuộc ba cạnh AB, AC, BD sao cho EF cắt BC tại I, AD cắt EG tại H. Chứng minh rằng ba đường thẳng CD, IG, HF cùng đi qua một điểm.

🐽Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho tứ diện ABCD. Gọi E, F, G lần lượt là các điểm thuộc ba cạnh AB, AC, BD sao cho EF cắt BC tại I, AD cắt EG tại H. Chứng minh rằng ba đường thẳng CD, IG, HF cùng đi qua một điểm.

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về chứng minh ba đường thẳng đồng quy để chứng minh ba đường thẳng đồng quy: + Gọi O là giao điểm của HF và IG + Chứng minh O thuộc CD.

Lời giải chi tiết

Gọi O là giao điểm của HF và IG. Ta có: \(O \in HF\), mà \(HF \subset \left( {ACD} \right) \Rightarrow O \in \left( {ACD} \right)\) Vì \(O \in IG\), mà \(IG \subset \left( {BCD} \right) \Rightarrow O \in \left( {BCD} \right)\) Do đó, \(O \in \left( {BCD} \right) \cap \left( {ACD} \right)\) Mặt khác, CD là giao tuyến của hai mặt phẳng (ACD) và (BCD) Do đó, \(O \in CD\). Vậy ba đường thẳng CD, IG, HF cùng đi qua một điểm.

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Tham Gia Group Dành Cho Lớp 11 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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