Ptolemaic inequalities (Q1919270)

If \(A_1 A_2 \dots A_n\) is a regular polygon with \(n\) odd, and \(P\) is any point on the arc \(A_n A_1\) of the circumcircle, then it is known that \[ |A_1 P|- |A_2 P |+ |A_3 P|- \dots - |A_{n - 1} P|+ |A_n P |= 0, \] where \(|A_i P|\) is the length of the chord \(A_i P\). The author shows that if \(P\) is a general point, then \[ |A_1 P|- |A_2 P|+ |A_3 P|- \dots - |A_{n-1} P|+ |A_n P|\geq 0, \] with equality if and only if \(P\) lies on the arc \(A_n A_1\). He also shows that the result is still true if each length is raised to the power \(d\), where \(d = 3,5,\dots,n-2\).