TITLE: "Classical" consequences of Bezout's theorem in dimensions $>2$
QUESTION [13 upvotes]: By Classical I mean something that could have been found before 1900 (say). 
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal's_theorem .
I am curious if there are some other statements that you find pretty that can be formulated (almost) as elementarily as Pascal's theorem and proven using higher dimensional Bezout's theorem? For example, is there some statement that involves quadrics, planes and lines (cubics?...)?
Motivation. I ask this question since I want to finish to teach my (introductory) course in algebraic geometry by higher-dimensional Bezout theorem (using Hilbert polynomials, ect), and I would be extremely happy to give some pretty application :). To give you an idea of the level of the course, it is based on some bits of Harris book "Algebraic geometry first course", 
Disclaimer. I don't doubt the usefulness of Bezout theorem and am sorry if the original question sounded like I doubt it. On the contrary I based the elementary course in algebraic geometry that I teach on this theorem. Namely, the course starts with Bezout for plane curves (using resultants), intorduces projective spaces and varieties, goes through Hilbert basis theorem and Hylbert polynomials (last section of Atiyah-Macdonald) and then as an applications we get a proof of a simplest version of Bezout's theorem in high dimension.
Also, It would be difficult for me to explain what I mean by pretty in math (for myself)
but still I feel that the using of this word is justified, because we, mathematicians use this word... Sometimes we disagree on what is pretty, but personally I find pretty huge amount of facts in algebraic geometry. In other words I will be happy to see any application that can be stated in the language on the level of my course.
In the comment I put the link to the question on stackexchange

REPLY [3 votes]: I just found one more classical consequence which is quite nice. It is described in the book of Harris "First course in algebraic geometry". 
Namely, one can prove that all projective automorphisms of $\mathbb CP^n$ are projective transformations. In order to prove this one shows that any hyperplane in $\mathbb CP^n$ is sent by any automorphism to a hyperplane (otherwise it would intersect the image of the line in more than one point). From this the statement follows rather directly.