Van Aubel's Theorem
Theorem
On each side of quadrilateral , construct an external square and its center: (, , , ; yielding centers ). Van Aubel's Theorem states that the two line segments connecting opposite centers are perpendicular and equal length: P_{AB}P_{CD} \perp P_{BC}P_{CD},
Proofs
Proof 1: Complex Numbers
Putting the diagram on the complex plane, let any point be represented by the complex number . Note that and that , and similarly for the other sides of the quadrilateral. Then we have
From this, we find that Similarly,
Finally, we have , which implies and , as desired.