Plane contains Infinite Number of Lines

Theorem
A plane contains an infinite number of distinct lines.

Proof
A plane contains an infinite number of points.

Not all these points are collinear.

Let $A$, $B$ and $C$ be points in a plane $P$.

From Propositions of Incidence: Line in Plane, any two of these points determine a line.

Consider the lines $AB$, $AC$ and $BC$, all of which are distinct.

Let $X$ be one of the infinite number of points on $BC$ which is not $B$ or $C$.

Then $AX$ is a line in $P$ which is distinct from both $AB$ and $AC$.

As there is are infinite number of points on $BC$, there are an infinite number of lines incident to $A$ and $BC$.

All these lines are in $P$.

Hence the result.