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.

What conjectures do you have?

Prove your conjecture(s).

Do your conjectures hold if the quadrilateral is not convex? Explore.

Suppose the Quadrilateral is a parallelogram. What additional conjectures and proofs can you find?

Extension
Consider the same questions for a quadrilateral formed by the centers of the four squares.