State the basic proportionality theorem. Use the theorem to do the following: In ABC, AD is the angle bisector of angle A. BA is produced to E such…

As DA \(\displaystyle \|\) CE \[\therefore \frac{\mathrm{BD}}{\mathrm{DC}}=\frac{\mathrm{BA}}{\mathrm{AE}} \quad \text {---- ① } \] \(\displaystyle \angle 2=\angle 3\) & \(\displaystyle \angle 1=\angle 4\)As \(\displaystyle \angle 1=\angle 2\) \[\therefore \angle 3=\angle 4 \] \(\displaystyle \Rightarrow \mathrm{AC}=\mathrm{AE}\) ---- ②From ① & ② \[\frac{\mathrm{BD}}{\mathrm{DC}}=\frac{\mathrm{BA}}{\mathrm{AC}} \]