Putnam 2003 B5

Let $A,B$, and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a, b, c$ be the distance from $P$ to $A, B, C$, respectively. Show that there is a triangle with side lengths $a, b, c$, and that the area of this triangle depends only on the distance from $P$ to $O$.