F.4 Maths -- trigonometry(II)

In triangle ABD
sin BAD / BD = sin B / AD
AD sin BAD = BD sin B ---------- ( i )

In triangle ACD
sin CAD / DC = sin C / AD
AD sin CAD = DC sin C ---------- ( ii )

Since BAD = CAD, from ( i ) and ( ii )
BD sin B = DC sin C
BD / DC = sin C / sin B ---------- ( iii )

In triangle ABC
sin B / AC = sin C / AB
sin C / sin B = AB / AC ---------- ( iv )

from ( iii ) and ( iv )
BD / DC = AB / AC