Let \(c\) be a real number satisfying \(c^2 \in {\mathbb Q}\), and let \(\phi_c=c/2+\sqrt{1+(c/2)^2}\). The authors show that \(\phi_c\) is a quotient of two diagonals of a regular polygon if and only if \[c \in \{0, \pm 1, \pm 3/2,\pm 2, \pm \sqrt{2}, \pm \sqrt{5}, \pm \sqrt{12}, \pm \sqrt{1/2}, \pm \sqrt{1/6}, \pm \sqrt{1/12}, \pm \sqrt{4/3}\}.\] In cyclotomic terms this condition can be expressed as \(\phi_c=|1-\zeta_N^a|/|1-\zeta_N^b|\), where \(\zeta_N\) is a primitive \(N\)th root of unity and \(a,b \in {\mathbb Z}\). Assuming that \(\gcd(a,b)=1\) they show that \([{\mathbb Q}(|1-\zeta_N^a|/|1-\zeta_N^b|): {\mathbb Q}]\) is \(\varphi(4N)/4\) for \(N\) odd, and at least \(\varphi(4N)/10\) for \(N\) even, where \(\varphi\) is Euler’s totient function. The proof is based on searching cyclotomic points for some families of polynomials.