# Proof about a projectivity of $\mathbb{P^2}$ and four lines of that projective space [closed]

Hello, someone could help me to solve this?

Let $r_1, r_2, r_3, r_4$ be four lines of $\mathbb{P^2}=P\mathbb{R^3}$ (projective space). Show that these two conditions are equivalent:

i) There exist a unique projectivity $\varphi$ of $\mathbb{P^2}$ whereby the four lines are invariants, but not necessarly composed by fixed points of $\varphi$.

ii) Each set of three of the four lines has empty intersection.

Eventually, generalize the proposition for n-dimension projective spaces.

Thank you

## closed as off-topic by user284331, Saad, Shailesh, Xander Henderson, Chris CusterApr 7 '18 at 3:07

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• What have tried so far? Where have got stuck? – Taroccoesbrocco Apr 6 '18 at 17:30
• To be honest I can't think a good idea about it – Arcticmonkey Apr 6 '18 at 19:34
• What is the representation of a projective line that you are used to ? – Jean Marie Apr 6 '18 at 19:46
• As an equivalence class of homogeneous coordinates – Arcticmonkey Apr 7 '18 at 6:02
• This definition only covers lines that are passing through the origin. Property ii) couldn't be fulfilled... – Jean Marie Apr 7 '18 at 13:10