A projective analogue of Napoleon's and Varignon's theorems (Q1649560)

The following classical results are well-known: -- Napoleon's theorem: For any triangle, the centers of the equilateral triangles erected on the outside of its edges form an equilateral triangle. -- Varignon's theorem: The midpoints of the edges of any quadrilateral form a parallelogram. The author of this paper shows that both theorems are in fact particular examples of a type of maps defined on the space of polygons and satisfying certain properties. Using this approach, the author also finds a new and analogous result in projective geometry. As it seems natural, the paper is closed with some open questions inviting the reader to explore the higher-dimensional setting or the case of non-Euclidean geometries.