A triangle theorem stating that an internal angle bisector from vertex A meeting side BC divides BC into segments proportional to the adjacent side lengths: if the bisector meets BC at D then BD/DC = AB/AC.

Principle

The organizing idea is that an angle bisector preserves ratios of opposing side lengths; equivalently, an internal bisector creates similar triangles that force proportional segments on the opposite side.

Demonstration

In triangle ABC let AB=6 and AC=9; the internal bisector of angle A meets BC at D, then BD/DC = 6/9 = 2/3. If BC has length 15, BD = (2/5)·15 = 6 and DC = 9, matching the side-length ratio.

Misapplication

Applying the internal-angle formula to an external angle bisector without sign changes; or using the theorem to conclude that any segment dividing BC in the ratio AB:AC must pass through A without verifying collinearity (the converse holds but requires checking).

Consequence

Permits exact construction of division points on a side from adjacent side lengths, supplies a direct method for length calculations in triangle geometry, and integrates with Ceva- and Menelaus-type ratio arguments.

Reversal

The external-angle bisector yields the external bisector theorem: the external bisector divides the opposite side externally in the ratio of the adjacent sides, i.e., BD/DC = -AB/AC under directed-segment conventions; conversely, proportional division can characterize a bisector when collinearity is known.

Boundary

Valid for nondegenerate triangles and for internal bisectors; in degenerate or collinear vertex configurations the statement is undefined. The internal theorem differs from the external version and from more general line-division statements in polygons.

Semantic Tension

Competes with Ceva's theorem and Menelaus's theorem for ratio-based characterizations: angle bisector gives a local proportionality, while Ceva/ Menelaus treat concurrency or collinearity of cevians/ transversals with product-of-ratios conditions.

Synthesis

The Angle Bisector Theorem ties an angular halving at a vertex to a precise division of the opposite side: bisecting an angle enforces a side-segment ratio equal to adjacent side lengths, a fact used routinely in length computations and proportionality arguments.