Prize Problem

(Note prize was earned by Ricardo Guerrero)

This is a problem I encountered recently when doing some renovation work in my house. 

I cut a length of wooden door stop for a straight, vertical, door frame of inside height h"  The door stop was too long for the door frame and when fitted with the top and bottom ends touching the door frame, bent in a circular arc such that the maximum horizontal distance between the arc and the door frame was p".  How much did I have to cut off the end of the door stop to make it fit exactly?  (Assume the door stop has zero thickness.)

The first complete and correct answer, showing details of working, submitted by a current, bona fide, COB student, will earn a prize of $10 and a mention on Dr. Banks’s web site.