Prove that a convex polygon $P$ can be partitioned into parallelograms if and only if it $P$ is centrally symmetric.
I can just verify the one side as follows:
Let $P$ be a polygon that can be dissected into parallelograms. Subdivide to assume that any two neighboring parallelograms share a full side. Start with one side. Then for any piece of that side, there is a well-defined path of parallelograms that ends when we get to another parallel section of the boundary of the same length. Since a convex polygon cannot have three parallel sides, this means we have another side parallel to the original of the same length. This argument shows that the sides of $P$ are parallel in pairs and congruent. This immediately implies that $P$ has an even number of sides, and by convexity, the opposite sides are the ones that are parallel and congruent. Hence P has a center of symmetry (the midpoint of the great diagonals).
I think we can do the other-side by induction. But I couldn't prove it.