If

is independent of the choice of

**proof**

Let **A,B,C,D** have normals **a,b,c,d** at O, and let P=[**p**]

Since P lies on all the p-lines, **p** is perpendicular to **a,b,c,d**,

so the normals lie on the plane through O perpendicular to **p**.

It follows that **a,b** span this plane, so there are real numbers

α,β,γ,δ such that

(*) **c**=α**a**+β**b**, **d**=γ**a**+δ**b**.

Note that **a**x**c**,**b**x**c**,**a**x**d**,**b**x**d**
are multiples of **p**, so we can

discuss their ratios. If we multiply the equations (*) by **c,d**,

we see that βγ/αδ = {**a**x**c**/**b**x**c**}/{**a**x**d**/**b**x**d**},
and so is

independent of the choice of **a,b,c,d**.

Let **L** have normal **u** at O. Then A=[**a**x**p**], etc.

If we multiply the equations (*) by [**p**], we see that

(A,B,C,D) = βγ/αδ, so is independent of the choice of **L**.