Given any quadrilateral, construct a square and locate its center on each side of the quadrilateral. Explore the relationship of the two segments defined by connecting the centers of squares on the opposite sides of the quadrilateral.
Explore with a GSP construction
Do your conjecture hold if the quadrilateral is not convex? Explore with a GSP construction and prove.
Suppose the quadrilateral is a parallelogram. What additional conjectures and proofs can you find? Consider also the quadrilateral formed by the centers of the four squares.
See Triangles and Squares for an investigation where the squares are constructed on the sides.