设CD=m ∠ADB=a
sin(∠ABC/2)=根号3/3 cos(∠ABC/2)=根号6/3
sin∠ABC=2sin(∠ABC/2)cos(∠ABC/2)=2根号2/3
cos∠ABC=1/3
由余弦定理 cosa=(4x^2+16/3-4)/(16√3m/3)
cos(180°-a)=-cosa=(x^2+16/3-BC^2)/(8√3m/3)
两式相加4x^2+16/3-4+2x^2+32/3-2BC^2=0
3x^2=BC^2-6 (1)
又由余弦定理AC^2=AB^2+BC^2-2AB*BCcos∠ABC
9x^2=4+BC^2-4BC*1/3
27x^2=12+3BC^2-4BC (2)
联立(1)(2) BC=15/2
三角形DBC的面积=(1/2)AB*BCsin∠ABC
=(1/2)*2*(15/2)*2√2/3
=5√2