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gbean

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**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!