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]