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. . . . . . .
A second theorem sometimes known as van Aubel's theorem states that if triangle A'B'C' is the Cevian triangle of a point P, then
AP/PA' = AB'/B'C + AC'/C'B
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.
Mathworld: van Aubel's Theorem describes both of van Aubel's theorems
Wikipedia: Van_Aubel's_theorem describes van Aubel's first theorem
Planet Math: Proof of Van Aubel's Theorem
Wolfram demonstrations: Van Aubel's Theorem for Quadrilaterals
Geometry Atlas: Finsler Hadwig Theorem
Cut-the-knot: Squares on Sides of a Quadrilateral contains a Java applet that lets you make your own examples to illustrate this theorem
Summary of geometrical theorems
Fundamental Theorem of Directly Similar Figures (FTDSF) -- if the lines connecting the corresponding vertices of two directly similar polygons are divided in equal ratios, then the resulting polygon is directly similar to the given two polygons.
The Finsler-Hadwiger Theorem -- Squares ABCD and PQRS joined at a corner D=P; the figure formed by joining the midpoints of AC, CQ, QS, and SA is a square. Special case of FTDSF. This factoid serves as the springboard for a proof of this factoid:
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.
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.
The webmaster and author of this Math Help site is Graeme McRae.