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In inversion,we extend the euclidean plane by adding a single point at infinity which lies on all the lines.But doesn't that mean lines can intersect twice?I mean non parallel lines already intersect once but then again the point at infinity lies on both of them,making $2$ intersections?Is this allowed?How do we see it intuitively?

a_i_r
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    If you added a point, then you should expect that some statements are satisfied by more points than before. You can think of the extended plane as an an sphere, with Stereographic projection. The north pole of the sphere being the new point that you added. The collection of circles and straight lines (clines) in the plane are sent to circles on the sphere. Circles on the sphere intersect at $0,1$ or $2$ points. – plop May 07 '21 at 11:05
  • Why should the north pole be the new point? – a_i_r May 07 '21 at 17:27
  • The choice made in the pictures in the link. – plop May 07 '21 at 17:39

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