Plane contains Infinite Number of Lines
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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.
$\blacksquare$
Sources
- 1952: T. Ewan Faulkner: Projective Geometry (2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence