Euclid's Elements - Book 1 - Proposition 31
Through a given point to draw a straight line parallel to a given straight line.
In other words, given a straight line (call it BC), and some point A not on the given line, construct a line through A that is parallel to BC.
Steps of the construction
1) Let A be the given point,
2) and BC the given straight line.
Note: We must construct a straight line, through point A, such that the line is parallel to line BC.
3) Let a point D be taken at random on BC,
4) and let AD be joined.
5) 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].
6) And let the straight line AF be produced in a straight line with EA.
Explanation: 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.
Therefore, through the given point A, the straight line EAF has been drawn parallel to the given straight line BC. █