desargues-3d-vector.md

Homogeneous coordinates can be denoted as 3D vectors. The line passing through two points, or the intersection point of two lines, can be determined by Cross Product. The collinearity and concurrency can be determined by Triple Product.

3D vector proof of Desargues's theorem

Denote points O, A₁, B₁, C₁ as O, A₁, B₁, C₁, then we have:

And we denote intersection ab as: ^[1]

Analogously, we have ac:

and bc:

Note that triple products in G, H and I are scalars, so we can simplify them as:

Then we get:

by using .

Finally, by using , and , we get:

which means ab, ac and bc are collinear.

3D vector proof of Pappus's Theorem

Given 4 arbitrary points A, B, C and D, from which no three points are collinear, we can denote D as:

E is collinear with A and C:

F is collinear with B and D:

Now let's calculate G:

And denote the triple product as :

Analogously, we have:

and

Then we get:

Finally, we get:

which means G, H and J are collinear.

Note

1. Here we should use some vector formulas (copied from the first page in John David Jackson's Classical Electrodynamics).