X.27

To find medial straight lines commensurable in square only which contain a rational rectangle.

Proof

Let two rational straight lines A, B commensurable in square only be set out; let C be taken a mean proportional between A, B, VI. 13 and let it be contrived that, as A is to B, so is C to D. VI. 12

Then, since A, B are rational and commensurable in square only, the rectangle A, B, that is, the square on C VI.17, is medial. X. 21

Therefore C is medial. X. 21

And since, as A is to B, so is C to D, and A, B are commensurable in square only, therefore C, D are also commensurable in square only. X. 11

And C is medial; therefore D is also medial. X. 23, addition

Therefore C, D are medial and commensurable in square only.

I say that they also contain a rational rectangle.

For since, as A is to B, so is C to D, therefore, alternately, as A is to C, so is B to D. V. 16

But, as A is to C, so is C to B; therefore also, as C is to B, so is B to D; therefore the rectangle C, D is equal to the square on B.

But the square on B is rational; therefore the rectangle C, D is also rational.

Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. Q. E. D.