cho tam giác ABC có 3 góc nhọn AB<AC , đường cao AH (H thuộc Bc ), Lấy M là trung điểm của AC . Từ M kẻ MI vuông gọc vs AH tại I , kẻ MD vuông góc với CH tại D 1, Cm :MIHD là Hình chữ nhật 2, CM : tam Giác AMI =tam Giác MCD ,và tứ giác AMDI là HBH 3, Gọi O là giao điểm của AD và Mi , qua M kẻ đường thằng // với AD và cắt đường thẳng CD tại K .Cm : Tam giácOMH =tam Giác MOK ai giúp với ạ
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Gọi ( O;R ) , ( I ;r ) lần lượt là các đường tròn ngoại tiếp tam giác ABC, DEF
Tam giác ABC ~ Tam giác DEF ( vì \(\widehat{ABC}=\widehat{DEF};\widehat{BAC}=\widehat{EDF}\)) \(\Rightarrow\widehat{ABC}=\widehat{DEF}\)
\(\widehat{ACB},\widehat{DEF}\)nhọn nên \(\widehat{ACB}=\frac{1}{2}\widehat{AOB};\widehat{DEF}=\frac{1}{2}\widehat{DIE}\)( hệ quả góc nội tiếp )
\(\Rightarrow\widehat{AOB}=\widehat{DIE}\)
\(OA=OB\left(=R\right)\Rightarrow\Delta OAB\)cân tại O
\(ID=IE\left(=r\right)\Rightarrow\Delta IDE\)cân tại I
Do đó Tam giác OAB ~ Tam giác IDE \(\Rightarrow\frac{OA}{ID}=\frac{AB}{DE}\Rightarrow\frac{R}{r}=\frac{3DE}{DE}\)
\(\Rightarrow R=3r\) ( đpcm)
Gọi ( O; R ), ( I; R ) lần lượt là các đường tròn ngoại tiếp tam giác ABC, DEF
Tam giác ABC ~ Tam giác DEF ( vì \(\widehat{ABC}=\widehat{DEF;}\widehat{BAC}=\widehat{EDF}\) ) \(\Rightarrow\widehat{ABC}=\widehat{DEF}\)
\(\widehat{ABC}=\widehat{DEF}\)nhọn nên \(\widehat{ACB}=\frac{1}{2}\widehat{AOB};\widehat{DEF}=\frac{1}{2}\widehat{DIE}\)(hệ quả góc nội tiếp )
\(\Rightarrow\widehat{AOB}=\widehat{DIE}\)
\(OA=OA\left(=R\right)\Rightarrow\Delta OAB\)cân tại O
Do đó Tam giác OAB ~ Tam giác IDE\(\Rightarrow\frac{OA}{ID}=\frac{AB}{DE}\Rightarrow\frac{R}{r}=\frac{3DE}{DE}\)
\(\Rightarrow R=3r\left(đpcm\right)\)
Rất vui vì giúp đc bạn <3