Question

tam giác ABC có 3 góc nhọn, đường cao AD và BE cắt nhau tại H. biết  \(\dfrac{AH}{HD}\)=k. chứng minh tan B x tan C = 1+k

Answer 1

\(\widehat{AHE}=\widehat{C}\)( cùng phụ \(\widehat{DAC}\))

\(\Rightarrow\widehat{BHD}=\widehat{C}\)

\(\underrightarrow{cm}\Delta HBD\sim\Delta CAD\left(g.g\right)\)

\(\Rightarrow\dfrac{BD}{AD}=\dfrac{HD}{CD}\)\(\Leftrightarrow BD.CD=AD.HD\)

\(\dfrac{AH}{HD}=k\Leftrightarrow\dfrac{AD}{HD}=k+1\)

\(tanB.tanC=\dfrac{AD}{BD}.\dfrac{AD}{DC}=\dfrac{AD^2}{BD.DC}=\dfrac{AD^2}{AD.HD}=\dfrac{AD}{HD}=k+1\)

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