proof of bisectors theorem

Consider sines law in trianglesMathworldPlanetmath APB and APC.

On APB we have


and on APC we have


Combining the two relationMathworldPlanetmath gives


However, APB+CPA=180, and so sinAPB=sinCPA. Cancelling gives


which is the generalizationPlanetmathPlanetmath of the theorem. When AP is a bisectorMathworldPlanetmath, BAP=PAC and we can cancel further to obtain the bisector theorem.

Title proof of bisectors theoremPlanetmathPlanetmath
Canonical name ProofOfBisectorsTheorem1
Date of creation 2013-03-22 14:49:28
Last modified on 2013-03-22 14:49:28
Owner drini (3)
Last modified by drini (3)
Numerical id 5
Author drini (3)
Entry type Proof
Classification msc 51A05