$S_{ABH} = S_{ABM} - S_{AHM}$
$=\dfrac{1}{2}.AH.(BM - HM) \leq \dfrac{1}{2}AH.BM$
$\Rightarrow S_{ABH}$ lớn nhất $\Leftrightarrow S_{ABH} = \dfrac{1}{2}.AH.BM = S_{ABM}$
$\Leftrightarrow HM = 0 \Leftrightarrow H \equiv M$
$\Rightarrow AM\perp BC$
$\Rightarrow S_{ABH} = \dfrac{1}{2}BM.AM = \dfrac{a^2}{2}$