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

a_{1}

a_{2}

a_{3}

and

a_{4}

be vectors representing the sides (in this order) of an arbitrary quadrilateral

Q

, so that

a_{1} + a_{2} + a_{3} + a_{4} = 0

(the zero vector). Now let

d_{1}

and

d_{2}

be the vectors representing the diagonals of

Q

. We may choose these so that

d_{1} = a_{4} + a_{1}

and

d_{2} = a_{3} + a_{4}

. Prove that

a_{2}^{2} + a_{4}^{2} - a_{1}^{2} - a_{3}^{2} = 2 (a_{1} \cdot a_{3} - a_{2} \cdot a_{4}) .

(1)

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

2 d_{1} \cdot d_{2} = a_{2}^{2} + a_{4}^{2} - a_{1}^{2} - a_{3}^{2} .

(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