Quiz #3

In Young's geometry, whose axioms are:
  1. There exists at least one line.
  2. Every line of the geometry has exactly three points on it.
  3. Not all the points are on the same line.
  4. For each two distinct points, there exists exactly one line on both of them.
  5. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line (Playfair's Axiom).

Prove that two lines parallel to a third line are parallel to each other.

(Parallel lines are lines that do not intersect - i.e., have no common point)

Solution:

Let lines l and m both be parallel to line n. BWOC assume that l and m meet at a point, say P. Since P is on l and l is parallel to n, P can not be on n. This is a contradiction to axiom 5, since we have a point (P) which does not lie on a line (n), and through P there pass two lines (l and m) which are parallel to n. Therefore, l and m must be parallel.