I don’t think there is a way to do that but have to ask, why an n-gon?
Not 100% sure but it looks like in your image that you already have vertices aligned so it’s just a matter to select all, ctrl+t and then alt+j (triangulate and convert to quads) and you can add a loop cut. Could also work with curve first and then convert it to a mesh.
[ATTACH=CONFIG]208614[/ATTACH]
to convert it to a curve you would need to delete a whole side
Then you can convert to a curve (there’s a doubled vertice in your blend, so press W->Remove Double before) and play with the bevel setting in the Curve tool panel
Convert back into a mesh after that.
But it will not do much good to center of the spiral that would not be pointy anymore, you would need to that part manually if you wish it to be pointy again.