CreatePalette[Pane[Row[

    Button[Style[First[#], 48, Darker[Magenta]], 

       NotebookWrite[InputNotebook[], 

        ToBoxes[ToExpression[NotebookRead[InputNotebook[]]] /. 

                     {Polygon[x_] :> 

            With[{y = Last[#][x]}, Polygon[y] /; True]}], After];

       SelectionMove[InputNotebook[], Previous, Graphics]

       ] & /@ {

      {"\[FilledDiamond]", (# + RotateLeft[#])/2 &}, 

      {"\[FivePointedStar]", 

       Module[{c = Mean[#], mids = (# + RotateLeft[#])/2}, 

         mids = ((# + c)/2 &) /@ mids; 

                   Flatten[Transpose[{#, mids}], 1]] &}, 

      {"\[FilledSquare]", 

       Module[{c = Mean[#]}, ((# + c)/2) & /@ #] &}

      }, "    "], ImageMargins -> 10], 

  WindowTitle -> "Polygon Operations"];