Construction of Parallel Line
Theorem
Given a straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line.
In the words of Euclid:
- Through a given point to draw a straight line parallel to a given straight line.
(The Elements: Book $\text{I}$: Proposition $31$)
Construction
Let $A$ be the point, and let $BC$ be the infinite straight line.
Take a point $D$ at random on $BC$, and construct the segment $AD$.
Construct $\angle DAE$ equal to $\angle ADC$ on $AD$ at point $A$.
Extend $AE$ into an infinite straight line.
Then the line $AE$ is parallel to the given infinite straight line $BC$ through the given point $A$.
Proof
The transversal $AD$ cuts the lines $BC$ and $AE$ and makes $\angle DAE = \angle ADC$.
From Equal Alternate Angles implies Parallel Lines it follows that $EA \parallel BC$.
$\blacksquare$
Historical Note
This proof is Proposition $31$ of Book $\text{I}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions