Question 45
Let acute triangle $A B C$ have circumcenter $O$, and let $M$ be the midpoint of $B C$. Let $P$ be the unique point such that $\angle B A P = \angle C A M$, $\angle C A P = \angle B A M$, and $\angle A P O = 90^\circ$. If $A O = 53$, $O M = 28$, and $A M = 75$, compute the perimeter of $\triangle B P C$.