



 
I get 12 also. The easiest way to see it seems to be to colour the original edges of the polygon in red, and the inner lines in blue. Once this is figured out, it is easy to derive a formula, by starting from a redredblue triangle and building up to the polygon.
All that remains is to label the vertices clockwise around the original labeled vertex, and perhaps stretch the polygon if you want it to be regular. There are $6 \times 2 \times 2 = 24$ ways to build a triangled polygon in this way. But each polygon has been counted twice, because there were two choices for the original redredblue triangle. So there are indeed 24/2 = 12 ways do divide the polygon, as you found out. This can be generalised to any number of sides. Claude  


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