BD^2 = BC^2-DC^2
BD = \sqrt{a^2-(\dfrac{1}{2}a )} = \dfrac{\sqrt{3} }{2} a
\dfrac{EF}{DC} = \dfrac{EB}{BD}
\dfrac{EF}{\dfrac{1}{2}a } = \dfrac{\dfrac{1}{2}a}{ \dfrac{\sqrt{3} }{2} a}
EF = \dfrac{\sqrt{3} }{6}a
FG^2 = EF^2-( \dfrac{1}{2}DE )^2 = \dfrac{1}{12}a^2 - \dfrac{1}{16}a^2
FG = \dfrac{\sqrt{3} }{12} a
S_{△DEF} = \dfrac{1}{2}DE\cdotp FG = \dfrac{1}{2}⋅\dfrac{1}{2}a⋅ \dfrac{\sqrt{3} }{12} a = \dfrac{\sqrt{3} }{48}a^2
S_{△ABC} = \dfrac{1}{2} DB⋅AC =\dfrac{1}{2} ⋅ \dfrac{\sqrt{3} }{2} a ⋅ a = \dfrac{\sqrt{3} }{4}a^2
\dfrac{S_{△ABC} }{S_{△DEF}} =12
S_{△DEF} = \dfrac{900}{12} =75