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State the basic proportionality theorem. Use the theorem to do the following : In $\Delta ABC$, AD is the angle bisector of angle A. BA is produced to E such that CE $||$ AD. Prove that $\frac{BD}{DC} = \frac{BA}{AC}$.
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Correct statement of Basic Proportionality Theorem.
As DA $||$ CE $\implies \frac{BD}{DC} = \frac{BA}{AE}$ --- (1)
$\angle 2 = \angle 3$ & $\angle 1 = \angle 4$. As $\angle 1 = \angle 2 \implies \angle 3 = \angle 4 \implies AC = AE$ --- (2)
From (1) & (2), $\frac{BD}{DC} = \frac{BA}{AC}$
As DA $||$ CE $\implies \frac{BD}{DC} = \frac{BA}{AE}$ --- (1)
$\angle 2 = \angle 3$ & $\angle 1 = \angle 4$. As $\angle 1 = \angle 2 \implies \angle 3 = \angle 4 \implies AC = AE$ --- (2)
From (1) & (2), $\frac{BD}{DC} = \frac{BA}{AC}$