Angle bisector divides opposite side in ratio of adjacent sides

Dependencies: Basic proportionality (intercept theorem) (BPT), Parallel ⇒ corresponding angles equal, Parallel ⇒ alternate angles equal, Isoceles converse: equal base angles ⇒ equal sides, Through a point off a line, exactly one parallel exists (Playfair).

Statement

Let $\triangle ABC$, and let $AD$ be the interior angle bisector of $\angle BAC$, with $D$ on the side $BC$ (i.e. $\angle BAD = \angle DAC$, $D \in \overline{BC}$). Then

In other words: the interior angle bisector divides the opposite side in the ratio of the two adjacent sides.