In triangle ABC, AD is the internal bisector of angle A.

by using angular bisector theorem in triangle ABC

(BD/DC) = (AB/AC) ----- (1)

In triangle ABC, AE is the internal bisector of angle A

(BE/CE) = (AB/AC) ----- (2)

from (1) and (2) we get,

(BD/DC) = (BE/CE)

(BD/BE) = (DC/CE)

Hence proved

(14) ABCD is a quadrilateral with AB = AD. If AE and AF are internal
bisectors of ∠BAC and ∠DAC respectively,then prove that the sides EF and
BD are parallel.

Solution:

In triangle ABC,AE is the internal bisector of ∠BAC

by using bisector theorem, we get

(AB/AC) = (BE/EC) ----- (1)

In triangle ADC,AF is the internal bisector of ∠DAC

by using bisector theorem, we get

(AD/AC) = (DF/FC)

Since lengths of AD and AB are equal, we are going to replace AB instead of AD

(AB/AC) = (DF/FC) ----- (2)

from (1) and (2) we get

(DF/FC) = (BE/EC)

So we can decide EF and BD are parallel by using converse of "Thales theorem".