Given a polygon, connect the midpoints of the sides, in order, to create a new polygon. We'll call this the mediogon of the original polygon. There's a theorem that every mediogon of a quadrilateral is a parallelogram. Proof: opposing sides of the mediogon are parallel to a common diagonal of the original quadrilateral.
The generalization beyond quadrilaterals is: which n-gons are mediogons of some other n-gon? Using vector algebra, the following are very easy to prove:
If n is even, then not all n-gons are mediogons (the class of mediogons can be described in a simple way), but each n-gon that is a mediogon is the mediogon of infinitely many n-gons.
If n is odd, then every n-gon is a mediogon of a unique other n-gon.
The problem is, every time I said "n-gon" above, I'm allowing the possibility that the edges might cross, so it's not a simple n-gon.
Which n-gons are mediogons of some simple n-gon?
My conjecture is that for even n, since every mediogon is a mediogon of infinitely many polygons, one of those polygons is always simple. For odd n, some subset of n-gons must be mediogons of simple n-gons.
The problem is that I don't know any workable characterization of simple polygons. I came up with a rule for when two line segments cross, it's a pair of big cumbersome inequalities involving inner products. I don't think I can apply it to this problem.