Through a given point to draw a straight line parallel to a given straight line.
Let A be the given point, and
BC the given straight
thus it is required to draw through the point A a straight line parallel to the straight line BC.
Let a point D be taken at random on BC, and let AD be joined; on the straight line DA, and at the point A on it, let the angle DAE be constructed equal to the angle ADC [I. 23] ; and let the straight line AF be produced in a straight line with EA.
Then, since the straight line AD
falling on the two straight lines
has made the alternate angles
equal to one another,
therefore EAF is parallel to BC. [I. 27]
Therefore through the given point A the straight line EAF has been drawn parallel to the given straight line BC. Q.E.F.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
Next: Proposition 32
Previous: Proposition 30
This proposition in other editions: