ABC

S

CE

BD

O

ADOE

BC=a

AC=b

\frac{b(3a+b)S}{2(a+b)(2a+b)}

\frac{S_{\triangle BEO}}{S_{\triangle BAD}}=\frac{BE}{BA}\cdot\frac{BO}{BD}.

\frac{BO}{OD}=\frac{BC}{CD}=\frac{2a}{b}

\frac{BO}{BD}=\frac{2a}{2a+b}

\frac{BE}{EA}=\frac{a}{b}

\frac{BE}{AB}=\frac{a}{a+b}

S_{\triangle BOE}=\frac{BO}{BD}\cdot\frac{BE}{BA}S_{\triangle ABD}=\frac{2a}{2a+b}\cdot\frac{a}{a+b}\cdot\frac{1}{2}S=a^{2}\cdot\frac{S}{(2a+b)(a+b)},

S_{ADOE}=\frac{1}{2}S-\frac{a^{2}S}{(2a+b)(a+b)}=\frac{1}{2}S\left(1-\frac{2a^{2}}{(2a+b)(a+b)}\right)=\frac{b(3a+b)S}{2(2a+b)(a+b)}.