geometry
what is minimum number of points in affine plane,
By the way: Here are the $\textbf{Three Axioms}$ for affine plane.
Given two distinct points $\textbf{P}$ and $\textbf{Q}$, there is only one line passing through them
Given a point $\textbf{P}$ and a line $\textit{l}$, if $\textbf{P}\not\in\textit{l}$,
there is only one line passing through point $\textbf{P}$ and parallel to line $\textit{l}$
There exist three points $\textbf{P}$, $\textbf{Q}$, $\textbf{R}$ non-collinear
Suppose by contradiction there exists a line $a$ passing through a single point $A$. By axiom 3 there exist two other points $B$ and $C$ and by axiom 1 they belong to a line $r$ parallel to $a$.
By axiom 1 there exists a line $s$ passing through $A$ and $B$ and by axiom 2 there exists a line $t$ passing through $C$ and parallel to $s$.
Line $t$ does not pass through $A$, so $t$ is parallel to $a$. But then we have two different lines $r$ and $t$ passing through $C$ and parallel to $a$, which contradicts axiom 2.
Two points determine a line (shown in the center). There are infinitely many infinite planes that contain that line. Only one plane passes through a point not collinear with the original two points:
Best Answer
HINT: If $K$ is any field, the vector space $V=K^2$ has always the structure of affine plane.
Now take for $K$ the smallest field you know.