# Axiomatic Euclidean geometry (Affine Geometry)

Geometry Level 5

Let $$(A,L)$$ be a pair of sets with $$A \neq \emptyset$$, where the elemets of $$A$$ will be called points, and $$L$$ is a set of subsets of $$A$$ ($$L \subseteq \mathbb{P}(A)$$) where elements of $$L$$ will be called lines.

Then $$(A,L)$$ is said to be an affine (euclidean) plane if it fulfills the 4 next axioms:

(Axiom I) Each element of $$L$$ has at least 2 points,i.e, each line has at least 2 points.(Actually, this is not an axiom, it can be deducted of the rest of following axioms.I have just written it as an axiom to expedite the solution. Without this assumption may be a long way until a complete solution)

(Axiom II) There exists at least 3 points in $$A$$ not aligned, i.es, there exists at least 3 points in $$A$$ which are not contained in any line.

(Axiom III) For all two distinct points given $$a, b \in A$$ there exists one and only one element $$l\ \in L$$ such that $$a,b \in l$$,i.e, for all two distinct points given there is one and only one line that contains both,

(Axiom IV. Euclides's Postulate) Given a line $$l \in L$$ and a point $$a \in A$$ , there exists one and only one line $$l'$$ parallel to $$l$$ which contains to $$a$$. (Two lines are parallel if they are equals or their intersection is the empty set).

Then, if $$(A,L)$$ is an affine (euclidean) plane, which is the minimum number possible of points in $$A$$?

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