Proving the Existence of Three Non-Collinear Points in an Affine Plane

[gbean]

Affine Plane--Geometry

Homework Statement

A = (P, L, I) is an affine plane. Prove that P (the set of all points) contains 3 distinct points that do not lie on a line.

Homework Equations

Can only use the following:
AP1: For any two distinct points P & Q, there exists one unique line incident (crosses through) with P and Q.
AP3: Each line is incident (crosses through) with at least 2 points.
AP4: There exist at least 2 lines.

The Attempt at a Solution

I'm having a lot of trouble writing rigorous proofs, so any help would be appreciated.

I used AP4 to say that there exist two lines, called l and m.
There are now two possible cases.
i) Either l is parallel/disjoint to m,
ii) or l intersects with m.

By AP3, each line l and m is incident (passes through) at least 2 points. l = L1L2, where L1 and L2 are points, and m = M1M2.

In case i), it isn't possible to draw a squiqqly sort of line that goes through any 3 points by AP1, which states that for any two distinct points, there is only 1 line that is incident with both points. So it is not possible to have 3 points on 1 line.

In case ii), define the intersection of l and m as a point n. Again, by AP1, for any 2 distinct points, there is only 1 line that is incident with both points. By this, there are 2 lines that contain 3 points.

I feel like I did not rigorously prove this, or that I am missing a few cases. Please help!

 

[HallsofIvy]

AP4 says that there exist at least 2 distinct lines, call them l1 and l2. Take, as part of your set of three points the two points that you know, by AP3, lie on l1. Since l2 is a different line from l1, there exist at least one point on l2 that is not on l1. Take that point as the third point in your set.

 

[gbean]

AP4 says that there exist at least 2 distinct lines, call them l1 and l2. Take, as part of your set of three points the two points that you know, by AP3, lie on l1. Since l2 is a different line from l1, there exist at least one point on l2 that is not on l1. Take that point as the third point in your set.

Would that be the entirety of a rigorous proof? Thank you very much for your help, I'm still trying to get the hang of this.