(* The corners of an n-gon... *)

poly[n_] := Table[{Cos[t], Sin[t]}, {t, 0, 1.999 Pi, 2 Pi/n}]

(* A test to see if two indexed corners can be opposites in an n-gon,
     in other words a test that they're not adjacent... *)

test[{i_, j_}, n_] := 1 < Mod[Abs[i - j], n] < n - 1;

(* All the possible pairs of opposite corners for an n-gon. *)

opposites[n_]:=
  Select[Union[
      Sort /@ Flatten[Outer[List, Range[n], Range[n]], 1]],
    test[#, n]&];

(* A silly trick to get patterns of the form

    {___,a,___,b,___} -> {a, b}. *)

myrule[n_] :=
  With[{patts = Table[Unique["e"], {n}]},
    Rule[Prepend[
        Flatten[Map[{Pattern[#, Blank[]], BlankNullSequence[]}&, patts]],
        BlankNullSequence[]], patts]];

(* For example...

   myrule[4]

   {___,e1_,___,e2_,___,e3_,___,e4_,___} -> {e1,e2,e3,e4} *)

(* A test to see if two diagonals of a polygon intersect... *)

test2[{___,{x_,y_},___,{a_,b_},___} /; (x<a<y<b || a<x<b<y)] := True;

(* Put it all together... Once you understand this, you'll be as bad at
     programming as I am. *)

showCatalan[n_]:=

  Module[{bg, corners, thelines, poly = N@poly[n]},
      bg = Line[Append[poly, First[poly]]];
> 
Transfer interrupted!