(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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,

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.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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