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Câu 15 trang 102 SGK Hình học 11 Nâng cao

Cho tứ diện ABCD. Tìm điểm O cách đều bốn đỉnh của tứ diện.
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Đề bài

Cho tứ diện ABCD. Tìm điểm O cách đều bốn đỉnh của tứ diện.

Lời giải chi tiết

Gọi I là tâm đường tròn ngoại tiếp của ΔBCDGọi d là đường thẳng đi qua I và vuông góc với mặt phẳng (BCD)Theo kết quả bài 14. M ϵ d ⇔ MB = MC = MD(d gọi là trục của đường tròn ngoại tiếp tam giác BCD)Gọi O là giao điểm của d với mặt phẳng trung trực của AB.=> OA = OB ( vì O thuộc mặt phẳng trung trực của AB).Và OB = OC = OD ( vì O thuộc đường thẳng d).Suy ra :OA = OB = OC = OD hay O cách đều bốn đỉnh của tứ diện (O gọi là tâm mặt cầu ngoại tiếp tứ diện ABCD).

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