Euclid's Elements Book II Proposition 13

To construct a square equal to a given rectilinear figure.

Let A be the

given rectilinear figure.

It is required to construct a square equal to the rectilinear figure A.

Construct the rectangular parallelogram BD equal to the rectilinear figure A. (I.45)

Then, if BE equals ED, then that which was proposed is done, for a square BD has been constructed equal to the rectilinear figure A.

But, if not, one of the straight lines BE or ED is greater.

Let BE be greater, and produce it to F.

Make EF equal to ED, and bisect BF at G.

Describe the semicircle BHF with center G and radius one of the straight lines GB or GF.

Produce DE to H, and join GH.

Then, since the straight line BF has been cut into equal segments at G and into unequal segments at E, the rectangle BE by EF together with the square on EG equals the square on GF.

But GF equals GH, therefore the rectangle BE by EF together with the square on GE equals the square on GH.

But the sum of the squares on HE and EG equals the square on GH, therefore the rectangle BE by EF together with the square on GE equals the sum of the squares on HE and EG.

Subtract the square on GE from each.

Therefore the remaining rectangle BE by EF equals the square on EH.

But the rectangle BE by EF is BD, for EF equals ED, therefore the parallelogram BD equals the square on HE.

And BD equals the rectilinear figure A.

Therefore the rectilinear figure A also equals the square which can be described on EH.

Therefore a square, namely that which can be described on EH, has been constructed equal to the given rectilinear figure A.

Q.E.D.

Proposition 13 <-- Proposition 14 --> Euclid's Elements: Book III