椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点F(-2,0),
∴a^=b^+4,①
椭圆过点(2,√6/3),
∴4/a^+(2/3)/b^=1.②
把①代入②*b^(b^+4),4b^+(2/3)(b^+4)=b^(b^+4),
b^4-(2/3)b^-8/3=0,
(b^-2)(b^+4/3)=0,b^=2,
代入①,a^=6,
∴椭圆方程是x^/6+y^/2=1.③
设l:x=my-3,④代入③*6,m^y^-6my+9+3y^=6,
(m^+3)y^-6my+3=0,
y1,2=[3m土√（6m^-9)]/(m^+3),
代入④,x1,2=[-9土m√(6m^-9)]/(m^+3),
看图,B(x2,y2),C(x1,-y1),
BF的斜率k1=y2/(x2+2)=[3m-√(6m^-9)]/[2m^-3-m√(6m^-9)],
CF的斜率k2=-y1/(x1+2)=[-3m-√(6m^-9)]/[2m^-3+m√(6m^-9)],
[3m-√(6m^-9)][2m^-3+m√(6m^-9)]-[-3m-√(6m^-9)][2m^-3-m√(6m^-9)]
=6m(2m^-3)+√(6m^-9)*[-2m√(6m^-9)]
=0,
∴k1=k2,
∴B、F、C三点共线.