Ta có:
$DE//AF$
$\to DE//AC$
$\Rightarrow ΔEBD \sim ΔABC$
$\Rightarrow \dfrac{S_{EBD}}{S_{ABC}} = \left(\dfrac{BD}{BC}\right)^2$
$\Rightarrow S_{EBD} = \dfrac{BD^2}{BC^2}S_{ABC}$
$\Rightarrow \dfrac{BD}{BC} = \sqrt{\dfrac{S_{EBD}}{S_{ABC}}}$
Tương tự, ta được:
$S_{FCD} = \dfrac{CD^2}{BC^2}S_{ABC}$
$\dfrac{CD}{BC} = \sqrt{\dfrac{S_{FCD}}{S_{ACB}}}$
Ta có:
$S_{AEDF} = S_{ABC} - S_{EBD} - S_{FCD}$
$= \dfrac{1}{2}AD.BC - \dfrac{BD^2}{BC^2}.\dfrac{1}{2}AD.BC - \dfrac{CD^2}{BC^2}\dfrac{1}{2}AD.BC$
$=\dfrac{1}{2}\cdot\dfrac{AD}{BC}(BC^2 - CD^2 - BD^2)$
$=\dfrac{1}{2}\cdot\dfrac{AD}{BC}.2BD.DC$
$= AD.BC.\dfrac{BD}{BC}\cdot\dfrac{DC}{BC}$
$= 2S_{ABC}.\sqrt{\dfrac{S_{EBD}}{S_{ABC}}}\cdot \sqrt{\dfrac{S_{FCD}}{S_{ACB}}}$
$= 2\sqrt{S_{EBD}}.\sqrt{S_{FCD}}$
$=2\sqrt{mn}$
