Angle Bisector Theorem

 In triangle ABC, angle bisector of A meets BC on D.
Then, BD/CD = AB/AC.

Proof:
Let Angle A = 2x.
Let Angle ADB = y and Angle ADC = 180 - y.

By Sine rule in triangle ABD:
BD/sin(x) = AB/sin(y)___________[1]

By Sine rule in triangle ACD:
CD/sin(x) = AC/sin(180-y) = AC/sin(y) (Since sin(y) = sin(180-y)). _________[2]

Using [1] and [2]:
sin(x)/sin(y) = BD/AB = CD/AC
=> AB/AC = BD/CD.



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