A Problem For Circle Lovers

A circle sliced by two parallel lines.Here’s a math problem for anyone who loves circles.

Have you ever noticed that if you slice a circle with two parallel lines, the two arcs contained between the lines always have the same length?

The problem is to show that this property almost (why almost?) characterizes circles:

Consider a simple closed curve that bounds a convex region in the plane. Suppose that any two sections of the curve that can be sandwiched between two parallel lines have equal length. Prove that the curve is a circle.

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