Sum of squares of chords to the points of a regular $n$-gon

The CSU math graduate students have started writing a really fun monthly math magazine for the department, featuring stories, mathematical tidbits, and puzzles collected from the department and slid under all the professors’ doors each month. It’s been a joy to read, and every month they have a “Problem of the Month’’ as well. This month’s problem was particularly fun: Let $P_1,P_2,\ldots,P_n$ be…