Okay, folks, time for another Problem of the Week! I hope it generates more response than last week’s problem:

Let $C$ be a simple closed curve in the plane, and let $P$ be any point strictly in the region interior to $C$. Show there are two points on $C$ whose midpoint is $P$.

Please submit solutions to topological[dot]musings[At]gmail[dot]com by Wednesday, July 9, 11:59 pm (UTC); do not submit solutions in Comments. Everyone with a correct solution will be inducted into our Hall of Fame! We look forward to your response.