Can Two Parallel Lines Actually Meet?

[irony of truth]

Hello.

I believe that there are proofs showing that two parallel lines meet at a certain point... but this cannot be proven through the concepts of a
simple geometry... some say that it can be proved by projective geometry.

Are there sites that show these proof?

 

[Data]

Parallel lines never intersect... in a Euclidean geometry. This is actually one of Euclid's postulates. It is not a necessary postulate, though, and you can find other geometries where parallel lines do intersect.

Here's some information:

http://mathworld.wolfram.com/ParallelPostulate.html

 

[HallsofIvy]

Parallel lines never intersect... in a Euclidean geometry. This is actually one of Euclid's postulates. It is not a necessary postulate, though, and you can find other geometries where parallel lines do intersect.

Here's some information:

http://mathworld.wolfram.com/ParallelPostulate.html

I think that's misleading. It is not that there exist geometries in which "parallel lines do intersect". Parallel lines, by definition, do not intersect. What is true is that there exist geometries (elliptic geometry) in which there are no parallel lines.

 

[TD]

When you work in projective geometry, isn't it so that 2 lines are said to intersect at a point at infinity?

 

[matt grime]

Yes, but still there are no *parallel* lines in projective geometry since all lines interesect.

 

[Data]

fair enough~