So BP3/BA=BO3/BD= 1/3, P3 is the bisector of AB.
bo4/Do4 = BP3/CD = BP3/BA = 1/3,
BP4/BA=BO4/BD= 1/4,
So P4 is the rendezvous point of AB,
Similar to provABle BPn/BA=BOn/BD= 1/n, so Pn is the n-bisector of ab.
(pay attention to the subscript, it's not easy to type)