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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