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Bolyai Theorem: Every pair of ulta-parallel lines has a unique line perpendicular to both lines (perpendicular transverse)

Does this imply that if there exists a line perpendicular to 2 other lines A and B, then A and B are parallel to each other?

Moreover, if we have 2 intersecting lines, then there doesn't exist a line perpendicular to both?

  • Intersecting lines with a common perpendicular would determine a triangle with two right angles, which is ... problematic. Thus, lines with a common perpendicular must be parallel. In fact, such lines must be "ultra-parallel" (aka, "divergently parallel"). If you're familiar with the Poincaré disk model, it's easy to see that, for "limiting parallel" lines/circles, the only candidate for a common perpendicular line/circle is the disk boundary itself, which doesn't count. – Blue Jun 18 '20 at 14:01
  • Ah right I get it, is there some sort of chat on stackexchange? I want to ask you some more questions on this topic –  Jun 18 '20 at 14:12

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