Angle bisector theorem

An internal angle bisector of a triangle divides the opposite side in the ratio of the two adjacent sides. If $AD$ bisects the angle at $A$ in triangle $ABC$, then $BD/DC=AB/AC$. Comparing the areas of triangles $ABD$ and $ACD$ proves this: their shared altitude gives the ratio $BD/DC$, while their equal included angles give $AB/AC$.