Two lines intersect at three points?

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Two lines, y=x and y=2x, can only intersect at one point in Euclidean geometry, specifically at the origin (0,0). The discussion highlights a misconception about finding intersection points at infinity, which are not considered valid in this context since infinity is not a real number. While the lines may appear to intersect at multiple points when considering infinite values, this is not applicable in standard geometry. The conversation also touches on alternative geometries, like spherical or hyperbolic, where lines may intersect in different ways. Ultimately, in Euclidean geometry, the lines intersect at only one point.
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Two lines intersect at three points?

I have a question that's been bugging my mind. :confused: Imagine the lines y=x and y=2x. Let's find their intersection point.
y=x and
y=2x so that
x=2x If we give x the value of 0 the equation will be correct. (1st intersection)
But if we give x the value of infinity the equation will be correct once again.
(inf=2*inf) (2nd intersection)
If we put negative infinity for x the equation will be correct. (3rd equation)
We have found three intersection points. We know that lines either do not intersect or intersect at only one point or intersect at infinite points. They certainly cannot intersect at three points. Is it because infinity is not a real number, which means that we don't care about the points of intersection whose coordinates are equal to either infinity or negative infinity? :wink:
 
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I assume x and y are real numbers? In which case you can not substitute in infinity because it is not a real number.
 
Mr. X said:
I have a question that's been bugging my mind. :confused: Imagine the lines y=x and y=2x. Let's find their intersection point.
y=x and
y=2x so that
x=2x If we give x the value of 0 the equation will be correct. (1st intersection)
But if we give x the value of infinity the equation will be correct once again.
(inf=2*inf) (2nd intersection)
If we put negative infinity for x the equation will be correct. (3rd equation)
We have found three intersection points. We know that lines either do not intersect or intersect at only one point or intersect at infinite points. They certainly cannot intersect at three points. Is it because infinity is not a real number, which means that we don't care about the points of intersection whose coordinates are equal to either infinity or negative infinity? :wink:

In Euclidean (planar) geometry, there are no points at infinity - so the two lines only meet at x=y=0.

There are other geometries where lines can meet in more than one place. For example, spherical or hyperbolic geometry.
 

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