Let $n$ be a positive integer. Suppose we have an equilateral polygon in the Euclidean plane with the property that all angles except possibly two consecutive ones are an integral multiple of $\frac{\pi}{n}$, then all angles are an integral multiple of $\frac{\pi}{n}$.
This problem is #28 on page 61 in these notes restated here for convenience: http://websites.math.leidenuniv.nl/algebra/ant.pdf
I have seen a number-theoretic proof of this. I was wondering if there are any geometric (or at least non number-theoretic) proofs of this result.