Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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Proposition 2.

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

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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; [Post. 3] 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.

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(Being) what it was required to do.

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