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;
let the straight line EG be bisected at H,
and let the straight lines CG, CH, CE be joined.
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 GHC is equal to the angle EHC.
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.
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.
We now concatenate the constructions.
1. Draw circle at C large enough.
Bisect GE.
Note: Two circles of equal distance, GE, and one of smaller distance, CE = CG.