Given a rectifiable simple closed curve, can one always find points P and Q on the curve that cut the curve into two arcs of equal length, neither of which crosses the chord joining P and Q? Here’s a proof that works if the curve is starlike from an interior point O: Every line through O determines a chord that cuts the curve into two pieces, one longer than the other (unless they’re the same length in which case we’re done). As you rotate the line by 180 degrees, the short piece becomes the long piece and vice versa. Since the lengths vary continuously, there must be some intermediate position of the line that makes the two pieces have equal length. For general Jordan curves, I can find either a proof nor a counterexample. (Seems like the kind of problem that Polish mathematicians in the 1920s would have looked at.) Jim Propp