An IMO Triangle Problem

The International Mathematical Olympiad (IMO) of 2006 was held in Slovenia. This Demonstration shows that

P

moves along the brown circle with center at the intersection of the circumcircle and the bisector of the angle

A

. The point

P

is constrained to move so that

∠PBA+∠PCA=∠PBC+∠PCB

. This is based on a problem presented at IMO as follows. Let ABC be a triangle with incentre

I

. A point

P

in the interior of the triangle satisfies

∠PBA+∠PCA = ∠PBC+∠PCB

. Show that

AP≥AI

, and that equality holds if and only if

P=I

.