simplify(solve({(BD + CD)^2 = AB^2 + AC^2 - 2*AB*AC*cos(Pi/9 + (4*Pi)/9), BD/CD = 4/3, AB*sin(Pi/9)/BD = AC*sin((4*Pi)/9)/CD, AB^2*CD + AC^2*BD = (BD + CD)*(4^2 + BD*CD), (AB, AC, BD, CD, AB + AC - (BD + CD), AC + (BD + CD) - AB, AB + (BD + CD) - AC) >~ 0}));

 gives AB = 28/3, but 

solve({BD^2 = AB^2 + 4^2 - 2*AB*4*cos(Pi/9), CD^2 = AC^2 + 4^2 - 2*AC*4*cos((4*Pi)/9), (BD + CD)^2 = AB^2 + AC^2 - 2*AB*AC*cos(Pi/9 + (4*Pi)/9), BD/CD = 4/3, (AB, AC, BD, CD, AB + AC - (BD + CD), AC + (BD + CD) - AB, AB + (BD + CD) - AC) >~ 0});

gives an error ...