We’ll actually prove the generalized angle “bisector” theorem since I find its proof more enlightening. However, it does require the law of sines.
Theorem
The goal is to prove that
Proof
From law of sines, we have that
Then, the key observation is that and are supplementary, and therefore their sines are equal. Cancelling from both sides yields the desired result.
Special cases
Note that if is a right angle, the theorem just collapses to the definition of sine.