Proposition 12
To draw a straight line perpendicular to a given infinite straight line from a given point not on it.

Let line AB be the given infinite straight line, and point C the given point which is not on it.
It is required to draw a straight line perpendicular to the given infinite straight line AB from the given point C which is not on it.

Take an arbitrary point D on the other side of the straight line AB, and describe the circle EFG with center C and radius CD. Bisect the straight line EG at point H, and join the straight lines CG, CH, and CE.

I say that line CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.

Since line GH equals line EH, and line CH is common, therefore the two sides line GH and line CH equal the two sides line EH and line CH respectively, and the base CG equals the base CE. Therefore the angle CHG equals the angle EHC, and they are adjacent angles.
But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Therefore line CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.