I was studying inversion in Olympiad geometry, and they (Evan Chen's book EGMO) mentioned that we can extend the Euclidean plane by adding a point $P_{\infty}$ such that each line passes through it, and no circle passes through it.
The reason for this they said was that now two parallel lines meet at that point only, and the center can go there on inversion.
But now I have a really stupid confusion:
Does this mean that all non parallel lines meet at two points and parallel lines meet at only one?
I am very confused by this part now, I tried looking up some things on Wikipedia but they had defined very different things and it just made me more confused.
I would really appreciate if anyone could clear this really dumb doubt of mine,
Thank you!
Best Answer
Yes, for any two distinct lines, that would be the case. You're trading the axiom that "two distinct points determine a line" for a different one where "three distinct points determine a 'cline'" (I thought they called them "lircles" actually...)
You are not doing Euclidean geometry anymore, but have passed to Möbius geometry. I suppose the chapter you are reading teaches you how to transition between the two.