Van Aubel's Theorem: Line segments connecting centers of squares on opposite sides of a quadrilateral are perpendicular and equal in length. This is true whether the squares are constructed on the outside or inside of the quadrilateral, and it is true even if the quadrilateral is complex (i.e. self-intersecting). The only requirement is that the squares be constructed in the same direction (clockwise or counterclockwise) taking each side of the quadrilateral in sequence.
Proof: see "my favorite party trick", below. . . . . . .
This is a generalization of the fact that the centroid of a triangle cuts each Cevian segment through it in the ratio 2:1.
My Favorite Party Trick, by Greg Muller has an excellent vector-based proof of van Aubel's theorem. It works by using vectors p and q each corresponding to half the length of two opposite sides of the quadrilateral, where p+q is the median vector. Likewise r and s are half the length of the other two opposite sides, and r+s is the other median vector. Then p' is p rotated 90 degrees to the right, and likewise q', r', and s'. Now p, q, r' and s' are perpendicular to p', q', r, and s, respectively. So r'+p+q+s' is perpendicular to p'+r+s+q', which connect the centers of the two squares that prove the theorem.
The figure formed by joining the side midpoints of a quadrilateral is a parallelogram is proved using vectors, along with the fact that the area of a quadrilateral is half the vector cross product of its diagonals. Moreover the area of the parallelogram is half that of the quadrilateral.