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Bài 3 trang 94 SGK Toán 11 tập 1 - Cánh diều

Cho ba đường thẳng a, b, c không cùng nằm trong một mặt phẳng và đôi một cắt nhau. Chứng minh rằng ba đường thẳng a, b, c cùng đi qua một điểm, hay còn gọi là ba đường thẳng đồng quy.

🌌Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

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

Cho ba đường thẳng a, b, c không cùng nằm trong một mặt phẳng và đôi một cắt nhau. Chứng minh rằng ba đường thẳng a, b, c𝄹 cùng đi qua một điểm, hay còn gọi là ba đường thẳng đồng quy.

Phương pháp giải - Xem chi tiết

Chứng minh giao điểm của ba đường thẳng nằm trên cùng một mặt phẳng, khi đó a, b, c cùng thuộc một mặt phẳng (trái với giả thiết) => giao điểm của ba đường thẳng phải trùng nhau.

Lời giải chi tiết

Giả sử a ∩ b = {I} và α = mp(a, b);

            a ∩ c = {J} và β = mp(a, c);

✨            b ∩ c = {K} và γ = mp(b, c) với các điểm I, J, K phân biệt.

🔯Khi đó α ∩ β = a và đường thẳng a chính là đường thẳng IJ.

🌺            α ∩ γ = b và đường thẳng b chính là đường thẳng IK.

♋            β ∩ γ = c và đường thẳng c chính là đường thẳng JK.

ไMà chỉ có một mặt phẳng duy nhất đi qua ba điểm I, J, K, đó là (IJK)

🉐Khi đó a, b, c cùng thuộc mặt phẳng (IJK), điều này trái với giả thiết a, b, c không cùng nằm trong một mặt phẳng.

Vậy I, J, K phải trùng nhau hay a, b, c đồng quy.

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

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