#### Proposition 12.

To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.

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

For let a point D be taken
at random on the other side of the straight line AB, and with centre C and distance CD let the circle EFG be described; [Post. 3]

let the straight line EG be bisected at H, [I. 10] and let the straight lines CG, CH, CE be joined. [Post. 1]

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

For, since GH is equal to HE, and HC is common,

the two sides GH, HC are equal to the two sides EH, HC respectively;
and the base CG is equal to the base CE;
therefore the angle CHG is equal to the angle EHC. [I. 8] And they are adjacent angles.

But, when a straight line set up 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. [Def. 10]

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

Q. E. F.

1 , . This is the full expression for a perpendicular, κάθετος meaning let fall or let down, so that the expression corresponds to our plumb-line. is however constantly used alone for a perpendicular, γραμμἡ being understood.

2 , literally “towards the other parts of the straight line AB,” AB. Cf. “on the same side” () in Post. 5 and “in both directions” () in Def. 23.