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Bài tập 10 trang 128 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC có ba đường cao AD, BE, CF . Biết AD = BE = CF. Chứng minh rằng tam giác ABC đều.
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Đề bài

Cho tam giác ABC có ba đường cao AD, BE, CF . Biết AD = BE = CF. Chứng minh rằng tam giác ABC đều.

Lời giải chi tiết

 

Xét ∆ACF (\(\widehat F = 90^\circ\)) và ∆ADC (\(\widehat D = 90^\circ\)) có AC (cạnh chung) và CF = AD (gt)Do đó: ∆ACF = ∆CAD (cạnh huyền – cạnh góc vuông).\( \Rightarrow \widehat {CAF} = \widehat {ACD}\) hay \(\widehat {CAB} = \widehat {ACB}\) (1)Xét ∆BEC (\(\widehat E = 90^\circ\)) và ∆BFC (\(\widehat F = 90^\circ\))Ta có: BC (cạnh chung)BE = CF (gt)Do đó: ∆BEC = ∆CFB (cạnh huyền – cạnh góc vuông).\( \Rightarrow \widehat {ACB} = \widehat {ABC}\) (2)Xét ∆BEA (\(\widehat E = 90^\circ\)) và ∆ABD (\(\widehat D = 90^\circ\)) có AB (cạnh chung) và BE = AD (gt)Do đó: ∆BEA = ∆ADB (cạnh huyền – cạnh góc vuông) \( \Rightarrow \widehat {BAC} = \widehat {ABC}\) (3)Từ (1), (2) và (3) \( \Rightarrow \widehat {CAB} = \widehat {ACB} = \widehat {ABC}\) => ∆ABC đều.

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