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.
Let the straight lines
be produced in a straight
line with DA,
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
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;
therefore AL is also equal to BC.
Therefore at the given point A
the straight line AL is
placed equal to the given straight
(Being) what it was required to do.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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