construct a parallel to a line through a point

I.31

Through a given point to draw a straight line parallel to a given straight line.

—Eulcid

Let A be the given point, and BC the given straight line; 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 BC, EF has made the alternate angles EAD, ADC 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.

references

[I.23]: /elem.1.23 “Book 1 - Proposition 23” [I.27]: /elem.1.27 “Book 1 - Proposition 27”