(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

To draw a straight line at right angles to a given straight line from a given point on it.

Let AB be the given straight line, and C the given point on it.

Thus it is required to draw from the point C a straight line at right angles to the straight line AB.

Let a point D be taken at random
on AC;

let CE be made equal
to CD;
[I. 3]

on DE let the equilateral
triangle FDE be
constructed,

and let FC be joined;

I say that the straight line FC
has been drawn at right angles to the given straight
line AB from
C the given point on it.

For, since DC is equal to
CE,

and CF is common,

the two sides DC,
CF are equal to the two sides
EC, CF
respectively;

and the base DF is equal
to the base FE;

therefore the angle DCF
is equal to the angle
ECF;
[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;
[Def. 10]

therefore each of the angles DCF,
FCE is right.

Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point C on it. Q.E.F.

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

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