Problem:
Determine the polygons with sides , not necessarily convex, which satisfy the property that the reflection of every vertex of the polygon with respect to every diagonal of the polygon does not fall outside the polygon.
Note: Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.
Solution:
A polygon with this property has to be convex, otherwise we consider an edge of the convex hull of this set of vertices which is not an edge of this polygon. All the other vertices are situated in one of the half-planes determined by the support-line of this edge, therefore the reflections of the other vertices fall outside the polygon.
Now we choose a diagonal. It divides the polygon into two parts, and . The reflection of falls into the interior of and vice versa. As a consequence, the diagonal is a symmetry axis for the polygon. Then every diagonal of the polygon bisects the angles of the polygon and this means that there are 4 vertices and the polygon is a rhombus. Each rhombus satisfies the desired condition.