(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

To place at a given point (as an extremity) a straight line equal to a given straight line.

Let A be the given point, and BC the given straight line.

Thus it is required to place at the point A (as an extremity) a straight line equal to the given straight line BC.

From the point A to the
point B let the straight line
AB be joined;
[Post. 1]

and on it let the equilateral
triangle DAB be constructed.
[I. 1]

Let the straight lines
AE, BF
be produced in a straight
line with DA,
DB;
[Post. 2]

with centre B and distance
BC let the circle
CGH be described;

and again, with centre D and
distance DG let the
circle GKL be described.
[Post. 3]

Then, since the point B is the centre of the circle CGH, BC is equal to BG.

Again, since the point D is the centre of the circle GKL, DL is equal to DG.

And in these DA is equal
to DB;

therefore the remainder AL
is equal to the
remainder BG.
[C.N. 3]

But BC was also proved
equal to BG;

therefore each of the straight
lines AL, BC is
equal to BG.

And things which are equal to the same thing are also equal
to one another;
[C.N. 1]

therefore AL is also
equal to BC.

Therefore at the given point A
the straight line AL is
placed equal to the given straight
line BC.

(Being) what it was required to do.

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

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