Moving Points
Projective!
Hopefully if you’re reading this you’re already comfortable (somewhat) with projective geometry. If not, you should learn about it. Basically everything on here is sourced from this paper, so I would highly recommend reading it before this. What I will write is a brief summary (and for some parts you will have to look at the pdf).
Method 1: Projective Maps
- A projective map is a function between two objects on which cross-ratio is defined which preserves cross-ratio e.g. inversion is a projective map
- If two maps coincide at 3 distinct points, they must be the same map (arises from the definition of cross-ratio)
- Therefore, in a problem, if you can characterised the given condition as a projective map, the statement-to-prove as another, and show they coincide at 3 distinct points, they must always coincide e.g. they must be the same one, proving the problem with just 3 cases!