angle bisector

Since ㄥOBA + ㄥOBC = 180°
So sin ㄥOBA = sin ㄥOBC (= K)
ㄥAOB = ㄥCOB (Given) . Hence sin ㄥAOB = sin ㄥCOB (= Q)
In △ AOB , by sine formula :
AB / sinㄥAOB = OA / sinㄥOBA
i.e. AB / Q = OA / K......(1)
In △COB , by sine formula :
BC / sin ㄥCOB = OC / sin ㄥOBC
i.e. BC / Q = OC / K......(2)
(1) / (2) :
(AB/Q) / (BC/Q) = (OA/K) / (OC/K)
AB : BC = OA : OC
(Proved)