C is a closed convex curve. If P lies on C and TP is the tangent at P, then TP varies continuously with P. Let O be a point inside C. Given a point P on C, define f(P) to be the point where the perpendicular from O to TP intersects C. Given P1, define the sequence Pn by Pn+1 = f(Pn). Assume that f is continuous and that, for each P, C lies entirely on one side of TP. Show that Pn converges. Find S = { P : P = limn→∞Pn for some P1}.