Question

If AS is the bisector of angle BAC AB=10 AC=14 BC=6 BD=?

Answer 1

Let D be the intersection of BC & AS.

As  AS bisects angle BAC,    angle DAB = angle DAC = A/2
In Δ DAB,      BD / sin A/2      =      AD / sin B    =    AB / sin (B+A/2)      --- eq 1
In Δ DAC,      CD / sin A/2      =      AD / sin C    =    AC / sin (B+A/2)      --  eq 2
we used        sin (180 - B - A/2)  = sin  (B+A/2)

Divide eq 1 by eq 2:  to get :
                   BD / CD  =    Sin C / sin B  =  AB / AC
     So BD/CD = 10/14  =>  add 1 on both sides,  ( BD+CD) / CD = 24/14
         BD + CD = BC = 6      => CD = 7/2    BD = 5/2