When Varignon meets Napoleon

Abstract In the well-known Napoleon’s theorem, equilateral triangles are built either all inwards or all outwards on the sides of a triangle. The centers of these triangles then form an equilateral triangle. Thébault’s theorem makes an analogous statement for parallelograms: If one erects squares on the sides (all inwards or all outwards), their centers form a square. Varignon’s theorem applies to general quadrilaterals: The centers of the sides form a parallelogram. We show the following for general quadrilaterals: Erect directly similar triangles on the sides, alternately inwards and outwards. Then the vertices, the centers of gravity, the orthocenters, the centers of the inscribed circle, or any other centers of these triangles always form a parallelogram. We also explore some of the nice geometric properties of these parallelograms.