Let the geometry prove it. Or, let Elliptic geometry prove the Euclidean problem.
With Euclidean plane geometry just a vast grid of points with empty space in between, there being 10^2412 points in the first quadrant with empty space of 10^-603 between successive Real points.
Now if we pick up a globe of Earth and take the grid system of longitude lines and latitudes we see that the empty space between intersection points is small near the poles but huge near the equator.
So in the Euclidean plane the empty spaces form perfect squares of 10^-603 sides but in the globe the empty spaces are spherical quadrilaterals that are small in area at the poles and get progressively larger towards the equator.
Now I bet mathematicians have figured out this size increase of those quadrilaterals at the poles as they grow larger towards to the equator. And I bet it is a factor of pi increase.
Now here is the thing to make the points of the grid of Euclidean geometry correspond directly to the points of the globe, a sphere of Elliptic Geometry. We have the same number of points in each, but we increase the size of the points as we approach the x-axis. Now we probably need all 4 quadrants. So if we did that in each of the 4 quadrants, that we are in fact transforming the Euclidean plane to be the surface of a sphere in Elliptic geometry.
So, now we see how a plane in flat Euclidean Geometry is transformed directly into a surface of Elliptic geometry, the sphere. And we can likely define Euclidean geometry as the geometry that has a uniform spacing of the empty space between successive Real points. Whereas Elliptic geometry has a variable size spacing depending on direction of a line (great circle).
Now as for the conjecture that the lines of Euclidean geometry given a single point and given all the angles possible that the line ray of an angle intersects with another point to be truly a line (two points determine a line)? Well, if Euclidean geometry is a 1-1 Correspondence and transformation of one to the other by varying the size of the empty space to form a sphere. We know the sphere lines (great circles) have a antipode to each podal point. And so each angle starting at the pole and encompassing all of the 360 degrees of angle possible, that each line has at least two points of intersection.
Quite simply, we let the geometry do the proof for us. And that is how it should be for most of mathematics, for if it is really true, then a proof is like a pleasure walk, not some arduous struggling workout with all sorts of questionable references.
Google's New-Newsgroups censors AP posts but Drexel's Math Forum does not and my posts in archive form is seen here: