(related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)

- Let the point $D$ have been taken a random on $BC$, and let $AD$ have been joined.
- And let (angle) $DAE$, equal to angle $ADC$, have been constructed on the straight line $DA$ at the point $A$ on it [Prop. 1.23].
- And let the straight line $AF$ have been produced in a straight line with $EA$.
- And since the straight line $AD$, (in) falling across the two straight lines $BC$ and $EF$, has made the alternate angles $EAD$ and $ADC$ equal to one another, $EAF$ is thus parallel to $BC$ [Prop. 1.27].
- Thus, the straight line $EAF$ has been drawn parallel to the given straight line $BC$, through the given point $A$.
- (Which is) the very thing it was required to do.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"