An IMO Triangle Problem

The International Mathematical Olympiad (IMO) of 2006 was held in Slovenia. This Demonstration shows that moves along the brown circle with center at the intersection of the circumcircle and the bisector of the angle . The point is constrained to move so that . This is based on a problem presented at IMO as follows. Let ABC be a triangle with incentre . A point in the interior of the triangle satisfies . Show that , and that equality holds if and only if .