[math-fun] Closed curve puzzle

In the plane: Let C be a C^oo simple closed curve. Let a "double-normal" be a line segment whose endpoints lie on C and which is normal to C at each of them. C must have a double-normal. (Proof: Consider the longest segment from C to C). Question: Let a "simple" double-normal be one that intersects C only at its endpoints. Must C have at least one simple double-normal? Prove or find a counterexample. --Dan _____________________________________________________________ * Thanks to Evan O'Dorney for this quick proof and for asking this question. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele