[ Properties of bisectors, concurrency ] [ The ratio in which the bisector divides the side $]$ [ Properties of bisectors, concurrency ]

In triangle $A B C$, points $M$ and $N$ are marked on sides $A B$ and $B C$ respectively, such that $B M=B N$. A line is drawn through point $M$ perpendicular to $B C$, and a line through point $N$ perpendicular to $A B$. These lines intersect at point $O$. The extension of segment $B O$ intersects side $A C$ at point $P$ and divides it into segments $A P=5$ and $P C=4$. Find the length of segment $B P$, given that $B C=6$.