Xét $ΔADB$ và $ΔAEC$ có:
$\widehat{D} = \widehat{E} = 90^o$
$\widehat{A}:$ góc chung
Do đó $ΔADB\sim ΔAEC\, (g.g)$
$\Rightarrow \dfrac{AD}{AE} = \dfrac{AB}{AC}$
$\Rightarrow \dfrac{AD}{AB} = \dfrac{AE}{AC}$
Xét $ΔADE$ và $ΔABC$ có:
$\dfrac{AD}{AB} = \dfrac{AE}{AC} \quad (cmt)$
$\widehat{A}:$ góc chung
Do đó $ΔADE\sim ΔABC \, (c.g.c)$
$\Rightarrow \dfrac{S_{ADE}}{S_{ABC}} = \left(\dfrac{AD}{AB}\right)^2 = \cos^2A$
$\Rightarrow S_{ADE} = S{ABC}.\cos^2A$