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Bài 4 trang 168 Tài liệu dạy – học Toán 7 tập 1

Giải bài tập Cho tam giác ABC cân tại A. Tia phân giác của góc B cắt tại F, tia phân giác của góc C cắt AB tại E.
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Đề bài

Cho tam giác ABC cân tại A. Tia phân giác của góc B cắt tại F, tia phân giác của góc C cắt AB tại E. a) Chứng minh rằng \(\widehat {ABF} = \widehat {ACE}.\) b) Chứng minh rằng tam giác AEF cân. c) Gọi I là giao điểm của BF và CE. Chứng minh rằng tam giác IBC và IEF  là những tam giác cân.

Lời giải chi tiết

 

a)Ta có: \(\widehat {FBC} = \widehat {ABF} = {{\widehat {ABC}} \over 2}\)   (BF là tia phân giác của góc ABC)\(\widehat {ECB} = \widehat {ACE} = {{\widehat {ACB}} \over 2}\)   (CE là tia phân giác của góc ACB)\(\widehat {ABC} = \widehat {ACE}(\Delta ABC\)  cân tại A)Do đó: \(\widehat {ABF} = \widehat {FBC} = \widehat {ECB} = \widehat {ACE} \Rightarrow \widehat {ABF} = \widehat {ACE}\)b)Xét tam giác AEC và AFB có:\(\widehat {EAC} = \widehat {FAB}\)   (góc chung)AC = AB (tam giác ABC cân tại A)\(\widehat {ACE} = \widehat {ABF}\)   (chứng minh câu a)Do đó: \(\Delta AEC = \Delta AFB(g.c.g) \Rightarrow AE = AF.\)   Vậy \(\Delta AEF\)  cân tại A.c) Ta có: \(\widehat {IBC} = \widehat {ICB}\)   (chứng minh câu a). Vậy tam giác IBC cân tại I.Ta có: \(\widehat {AEF} + \widehat {IEF} = \widehat {AEI};\widehat {AFE} + \widehat {IFE} = \widehat {AFI}\)Mà \(\widehat {AEF} = \widehat {AFE};\widehat {AEI} = \widehat {AFI} \Rightarrow \widehat {IEF} = \widehat {IFE}\)Do đó tam giác IEF cân tại I.

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