A convex quadrilateral Q is 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) 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.

As d₁=a₄ + a₁ and d₂ = a₃ + a₄ it follows that a₁ + a₂ = -d₂, a₂ + a₃ = -d₁.

a22+a42-a12-a32 =(a22-a12)+(a42- a32) =(a2-a1)( a2+a1)+(a4-a3)( a4+a3) =-d2(a2- a1)+d2(a4-a3) = d2(a4-a3-a2+a1) =d2((a4+a1)-(a3+ a2)) =d2(d1+d1) =2d2 . d1 .

Now a₁+a₂+a₃+a₄=0 implies that a₄=-a₁-a₂- a₃.

a1 ·a3-a2 . a4 = a1 . a3-a2(- a1-a2-a3) =a1 . a3+a2 . a1+a2. a2+a2. a3 = a1(a2+a3)+a2( a2+a3) =(a1+a2)( a3+a2) =(-d1)(-d2) =d1. d2 .

Hence

2(a1 . a3-a2 . a4) =2d1 . d2 = a22+a42-a12-a32 .

If the diagonals of Q are perpendicular in one position of Q, then 2d₁ . d₂=a₂²+a₄²-a₁²- a₃² = 0. As a₁,a₂, a₃,a₄ are constant in length a₂²+a₄²-a₁²-a₃² will always be zero which implies that d₁ . d₂=0, so they are perpendicular in all variations of Q.