Trong $∆ABC$ kẻ đường cao $CH$
$\Rightarrow \begin{cases}S_{AMC}=\dfrac12AM.CH\\S_{ABC}=\dfrac12AB.CH\end{cases}$
$\Rightarrow \dfrac{S_{AMC}}{S_{ABC}}=\dfrac{AM}{AB}$
$\Rightarrow S_{AMC}=\dfrac{AM}{AB}.S_{ABC}\qquad (1)$
Trong $∆AMC$ kẻ đường cao $MK$
$\Rightarrow \begin{cases}S_{AMN}=\dfrac12AN.MK\\S_{AMC}=\dfrac12AC.MK\end{cases}$
$\Rightarrow \dfrac{S_{AMN}}{S_{AMC}}=\dfrac{AN}{AC}$
$\Rightarrow S_{AMN}=\dfrac{AN}{AC}.S_{AMC}\qquad (2)$
$(1)(2)\Rightarrow S_{AMN}=\dfrac{AM.AN}{AB.AC}.S_{ABC}$
$\Rightarrow \dfrac{S_{AMN}}{S_{ABC}}=\dfrac{AM.AN}{AB.AC}\qquad (đpcm)$
