Matthew Hudelson, Formulas among diagonals in the regular polygon  and the Catalan numbers,
Forum Geometricorum, 6 (2006) 255--262.

Abstract: We look at relationships among the lengths of diagonals in the regular polygon.  Specifically, we derive formulas for all diagonals in terms of the shortest diagonals and other formulas in terms of  the next-to-shortest diagonals, assuming unit side length.  These formulas are independent of the number of sides of the regular polygon.  We also show that the formulas in terms of the shortest diagonals involve the famous Catalan numbers.

[ps file][pdf]