On the Existence of Non-Intersecting Straight Lines on the Plane

In this brief note, we adduce the logical rationale that if at least one infinite straight line non-intersecting with the given straight line passes through a given point not lying on a given straight line, then it must be unique.


Brief Historical Background
One axiom of Euclidean geometry had bothered mathematicians somewhat, not because there was in their minds any doubt of its truth but because of its formulation.We have in mind the parallel postulate or, as it is often referred to, Euclid's fifth postulate.As Euclid worded it, it states: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines if extended will meet on that side of the straight line on which the angles are less than two right angles [1].
Mathematicians regard that the parallel postulate in the form stated by Euclid was thought to be somewhat too complicated.Euclid apparently feared to assume that there could be infinite straight lines.Nevertheless Euclid did imply the existence of infinite straight lines for, were they finite, they could not be ex- tended as far as necessary in any given context.So hereafter we will use the existence of infinite straight lines at an infinite plane and allude to the received version of the parallel postulate proposed by John Playfair in 1795 which states: Through a given point n not on a straight line G (Figure 1) there is one and only one straight line N in the plane of n and G which does not intersect with G [2].In our work, we will try to avoid using the word-group "parallel straight lines" but will use the phrases "non-intersecting" and "intersecting" straight lines.

The Existence of a Straight Line That Does Not Intersect with Another Straight Line
Since we will use some common statement for the Euclidean and Lobachevsky geometries, let us now somewhat change the literal sense of the axiom of Playfair and try to prove the following statement: We assume that there exists at least one infinite straight line N in the plane of n and G, which passes through a given point  not lying on an infinite straight line G and does not intersect with G (i.e.N G∈∅  or N G   ) (Figure 1).In this case this infinite straight line is the only one passing through the point  that does not intersect with G.
We shall rely on the first postulate of Euclid (axiom of belonging).The following cases of mutual arrangement are possible for two arbitrary straight lines: they may not have common points at all, have one point of intersection or completely coincide.If two straight lines have two different common points, then these lines coincide (all points are common for them).
Admitting a hypothesis of the existence of infinite straight lines at an infinite plane, we adduce here the following reasoning.Let us move the infinite straight line I which intersects two a priory mutually non-intersecting infinite straight lines N and G at the points n and g, respectively (Figure 1), so that the point n will be remained stationary, while the point g would move to the right (for definiteness only; the opposite motion can be analyzed similarly).In order to straight lines I and G no longer intersect, their right ends must first be combined 1 then must be split.But, as it follows from the meaning of the word "infinite", these lines do not have ends, so, these non-existent ends cannot be combined and, accordingly, these straight lines cannot be split, that is cannot become disjoint (or non-intersecting)!Moreover, what happens to the angle α when the point g moves to the right?The angle α will tend to zero (instead of moving, we also can choose an infinite sequence of points g k having a limit point +∞; as a result, we obtain the limit point 0 k α → ).But the angle α cannot reach zero 1 We require here that the left (with respect to the point n) parts of these straight lines initially do not intersect.

Figure 1 .
Figure 1.The geometrical proof of the theorem for non-intersecting straight lines.