Line segments connecting centers of squares on sequential sides of a parallelogram form a square. This is a special case of van Aubel's Theorem.
The vertices of equilateral triangles constructed on the outside (or inside) of two consecutive sides of a square, together with the "untouched" vertex of the square form an equilateral triangle.
Wikipedia: Thebault's theorem
Summary of geometrical theorems
Van Aubel's Theorem: Line segments connecting centers of squares on opposite sides of a quadrilateral are perpendicular and equal in length. (See also van Aubel's second theorem.)
Napoleon's Theorem: if equilateral triangles are constructed on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves form an equilateral triangle.
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