Angle bisector theorem
= 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 $AB/AC$.