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