External angle bisector property

Dependencies: Basic proportionality (intercept theorem) (BPT), Parallel ⇒ corresponding angles equal, Parallel ⇒ alternate angles equal, Isoceles converse: equal base angles ⇒ equal sides, Linear pair sums to 180°.

Statement

Let $\triangle ABC$, and let $AD$ be the external angle bisector of $\angle A$ — that is, $AD$ bisects the exterior angle at $A$ (which is supplementary to the interior angle $\angle BAC$). When $|AB| \neq |AC|$, the line $AD$ meets the extension of the opposite side $BC$ at a unique point $D$, and

In other words: the external angle bisector divides the opposite side in the ratio of the adjacent sides "externally" — $D$ does not lie inside $\overline{BC}$ but on its extension. When $|AB| > |AC|$, $D$ lies beyond $C$; when $|AB| < |AC|$, $D$ lies beyond $B$. Degenerate case: when $|AB| = |AC|$, the external angle bisector is parallel to $BC$ and there is no intersection point; the theorem holds vacuously.