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So, I asked a question about how to find if three lines are concurrent. I built the algorithm I needed, and it was working well, until I started doubting my power of judgement. So my question:

Are two equal lines also intersecting?

For an example, $9x+9y=9$; $7x+7y+7$ ? Because now, if I would draw these lines and add any line to the comparison, these three lines would be concurrent.


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Yes; they intersect at all of their points. – Qiaochu Yuan Sep 23 '10 at 20:46
NOTE If you have 2 equal lines L, then it's not true that any 3rd line works since it won't intersect L if it's parallel to L. But once you have 2 equal lines L the problem reduces to a simpler one: if any other line intersects L then that gives your point of multiplicity 3; else all lines are parallel to L and you need only check if there are 3 equal lines. – Bill Dubuque Sep 23 '10 at 21:12
The answer to your question really depends on whether you want coincident lines to be intersecting or not! – Mariano Suárez-Alvarez Sep 24 '10 at 7:41
@Janis, then you should be asking him. – Mariano Suárez-Alvarez Sep 24 '10 at 8:25
But you'd b asking your teacher: by asking him questions (and no, this is not a stupid question) it is precisely that you are supposed to get that sort of background!!!! – Mariano Suárez-Alvarez Sep 24 '10 at 23:25
up vote 3 down vote accepted

In the analytical geometry version of Euclidean Geometry we need a convenient way of thinking of parallelism and there is also the interest in constructing the real projective plane (where there are no parallel lines) from Euclidean Geometry. (The key idea is that we think of all the lines which have the same slope as having a special additional point on them at "infinity" (thereby creating new lines in the projective plane) and all these new points, one for each slope, lie on a new line, the line at "infinity.") In the analytical geometry framework of Euclidean Geometry lines are parallel when they have the same slope but there is the problem that a nonzero multiple of an equation looks like a different equation but consists of all of the same points. Furthermore, one would like parallelism to be an equivalence relation, which means it obeys being reflexive, symmetric and transitive.

So it is CONVENIENT to think of a line as being parallel to itself. Now all lines with the same slope can be thought of as parallel. (All the lines with undefined slope are also parallel to each other; they are the vertical lines.) Furthermore, we can think of the Euclidean plane's lines as being partitioned in classes by this parallelism relation.

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