$\triangle$BDH $\backsim$ $\triangle$BAC (g.g)
$\rightarrow$ $\dfrac{S_{BDH}}{S_{ABC}}$=$\dfrac{BH^{2}}{BC^{2}}$
Tương tự $\dfrac{S_{CHE}}{S_{CBA}}$=$\dfrac{CH^{2}}{BC^{2}}$
$\Rightarrow$ $\dfrac{S_{BDH} + S_{CHE}}{S_{ABC}}$ =$\dfrac{BH^{2}+CH^{2}}{BC^{2}}$ $\ge$$\dfrac{\dfrac{(BH +CH)^{2}}{2}}{BC^{2}}$ = $\dfrac{1}{2}$
$\Rightarrow$ $\dfrac{S_{ADHE}}{S_{ABC}}$ $\le$$\dfrac{1}{2}$
$\Rightarrow$ $S_{ADHE}$ max =$\dfrac{S_{ABC}}{2}$
