Points of difference: Relative infinity in the Euclidean plane (Q1207030)

Let be given a Möbius plane \(M\). By distinguishing a single point \(e_ i\in M\) \((i=1,2;e_ 1\neq e_ 2)\) as his (her) ``point at infinity'', each of two ``observers'' \(O_ i\) \((O_ 1\) male, \(O_ 2\) female) is left with an Euclidean plane (up to measuring scale!) \(E_ i:=M\backslash e_ i\). The observer \(O_ i\) calls cycles ``lines'' if they contain \(e_ i\) and ``circles'' otherwise; he (she) measures the angle between two rays (ray := section of a cycle between some initial point and \(e_ i)\) with the same initial point to be the angle between the cycles containing them. Obviously, both observers will disagree over the distinction between lines and circles except for common lines, which contain both ``antipodes'' \(e_ 1\), \(e_ 2\), and common circles, which contain neither. The author now reveals many further areas of agreement and disagreement, mainly concerning ``angles and triangles'' (\S3) and ``mutually inscribed triangles'' (\S4). Examples: The observers \(O_ 1,O_ 2\) agree on which cycle is the circumcircle of a given triangle, but disagree on its centre; if \(\Delta_ 1\), \(\Delta_ 2\) are mutually inscribed triangles, then \(e_ 2\) is the orthocentre of \(\Delta_ 1\) iff \(e_ 1\) is the orthocentre of \(\Delta_ 2\) (theorem 4.4). \S5 deals with the examination how \(O_ 1,O_ 2\) relate distances and areas when they are allowed measuring scales. In \S6, some questions suggest directions for further investigations of the effects of a relative infinity.