Consider a convex quadrilateral Q made from four rigid rods with flexible joints at the vertices so that the shape of Q can be changed while keeping the lengths of the sides constant. Let a₁, a₂, a₃ and a₄ be vectors representing the sides (in this order) of an arbitrary quadrilateral Q, so that a₁+a₂+a₃+a₄ = 0 (the zero vector). Now let d₁ and d₂ be the vectors representing the diagonals of Q. We may choose these so that d₁=a₄+a₁ and d₂=a₃+a₄. Prove that

a₂²+a₄²−a₁²−a₃² = 2(a₁·a₃−a₂·a₄).

(1)

and that the scalar product of the diagonals is constant and given by:

2d₁·d₂ = a₂²+a₄²−a₁²−a₃².

(2)

Use these results to show that, as the shape of the quadrilateral is changed, if the diagonals of Q are perpendicular in one position of Q, then they are perpendicular in all variations of Q.