a) Xét \(\Delta HAB,\Delta MAH\) có :
\(\left\{{}\begin{matrix}\widehat{A}:Chung\\\widehat{AHB}=\widehat{AMH}=90^o\end{matrix}\right.\)
\(\Rightarrow\Delta HAB\sim\Delta MAH\left(g.g\right)\)
Xét \(\Delta HAC,\Delta NAH\) có :
\(\left\{{}\begin{matrix}\widehat{A}:chung\\\widehat{AHC}=\widehat{ANH}=90^o\end{matrix}\right.\)
=> \(\Delta HAC\sim\Delta NAH\left(g.g\right)\)
b) Từ \(\Delta HAB\sim\Delta MAH\left(g.g\right)\) ta có :
\(\dfrac{AM}{AH}=\dfrac{AH}{AB}\)
\(\Rightarrow AM.AB=AH^2\) (1)
Từ \(\Delta HAC\sim\Delta NAH\left(g.g\right)\) ta có :
\(\dfrac{AN}{AH}=\dfrac{AH}{AC}\)
\(\Rightarrow AN.AC=AH^2\) (2)
Từ (1) và (2) suy ra : \(AM.AB=AN.AC\left(=AH^2\right)\)
c) Xét \(\Delta AMN,\Delta ACB\) có :
\(\left\{{}\begin{matrix}AM.AB=AN.AC\left(cmt\right)\Leftrightarrow\dfrac{AM}{AN}=\dfrac{AC}{AB}\\\widehat{A}:Chung\end{matrix}\right.\)
\(\Rightarrow\Delta AMN\sim\Delta ACB\left(c.g.c\right)\)