Let's start by defining the cross ratio more precisely.
Are there any other sets of collinear points for which we might want to check the cross ratio? How about four evenly spaced points?
In case you've encountered the cross ratio elsewhere, you might have seen it defined differently. This is because there's actually several different formulations of the cross ratio, all equally valid!
Let's prove the invariance of the cross ratio -- and finally answer the question of what stays the same under a perspective shift!
In addition to its invariance under perspective shifts, the cross ratio displays another very important property.
Putting together the invariance and injectivity of the cross ratio, we can finally prove the Three Fixed Points Theorem, which has been at the heart of all our major results!
As we saw while proving the injectivity theorem, it's often natural to fix three points and think of the cross ratio as a function on the fourth. Let's finish our introduction to the cross ratio by taking a closer look at that function.