Cho tam giác ABC có 3 góc nhọn (AB<AC). Các đường cao AK, BE, CF cắt nhau tại H. a) Chứng minh tam giác ABK đồng dạng với tam giác CBF. b) Chứng minh: CH*CF=CE*CA .c)Gọi N là giao điểm của AK và EF; D là giao điểm của đường thẳng BC và EF, gọi O và I lần lượt là trung điểm của BC và AH. Chứng minh ON vuông góc với DI
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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
a: Xét ΔBKA vuông tại K và ΔBFC vuông tại F có
\(\widehat{KBA}\) chung
Do đó: ΔBKA~ΔBFC
b: Xét ΔCEH vuông tại E và ΔCFA vuông tại F có
\(\widehat{ECH}\) chung
Do đó: ΔCEH~ΔCFA
=>\(\dfrac{CE}{CF}=\dfrac{CH}{CA}\)
=>\(CE\cdot CA=CH\cdot CF\)