Angle bisector theorem

Here AD is angle bisector of angle A

Sol:-

First of all draw perpendicular BE & CF to AD

Now in ∆ABE &  ∆ ACF :-

Angle 1= Angle 2            (As AD is Angle bisector)

Angle AEB = Angle AFC  ( 90 DEGREE)

So Angle 3 = Angle 4

Hence both triangle are similar

AB/AC = BE/CF                     Eqn.  (1)

Now in ∆BED &  ∆ CFD :-

Angle 5 = Angle 6           (Vertically opposite angle)

Angle 7 = Angle 8         ( 90 DEGREE)

Hence both triangle are similar

BE/CF = BD/CD                  Eqn.  (2)

 Comparing equation 1 &2 :-

AB/AC = BD/CD                  (H.P)