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 $H$, [Prop. 1.10] 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 $EHC$. [Prop. 1.8] 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.