a)
$EM=\sqrt{AM.BM}=\sqrt{6,4.3,6}=4,8cm$
${{S}_{\Delta ABE}}=\dfrac{1}{2}EM.AB=\dfrac{1}{2}\cdot 4,8\cdot 10=24c{{m}^{2}}$
b)
Có $\begin{cases}AE^2=AM.AB\\AE^2=AN.AC\end{cases}$
$\Rightarrow AM.AB=AN.AC$
$\Rightarrow \Delta AMN\backsim\Delta ACB\left( c.g.c \right)$
c)
Có $AM.AB=A{{E}^{2}}$
$\Rightarrow \dfrac{AM}{AB}=\dfrac{A{{E}^{2}}}{A{{B}^{2}}}={{\sin }^{2}}B$
d)
Có $AN.AC=A{{E}^{2}}$
$\Rightarrow \dfrac{AN}{AC}=\dfrac{A{{E}^{2}}}{A{{C}^{2}}}={{\sin }^{2}}C$
Vì $\Delta AMN\backsim\Delta ACB$
$\Rightarrow \dfrac{{{S}_{\Delta AMN}}}{{{S}_{\Delta ACB}}}=\dfrac{AM}{AC}\cdot \dfrac{AN}{AB}=\dfrac{AM}{AB}\cdot \dfrac{AN}{AC}={{\sin }^{2}}B.{{\sin }^{2}}C$