| To find medial straight lines commensurable in square only which contain a rational rectangle. | ||
| Set out two rational straight lines A and B commensurable in square only. Take a mean proportional C between A and B. Let it be contrived that A is to B as C is to D. | X.10
VI.13 VI.12 | |
| Then, since A and B are rational and commensurable in square only, therefore the rectangle A by B, that is, the square on C, is medial. Therefore C is medial. | VI.17
X.21 | |
| And since A is to B as C is to D, and A and B are commensurable in square only, therefore C and D are also commensurable in square only. | X.11 | |
| And C is medial, therefore D is also medial. | X.23.Note | |
| Therefore C and D are medial and commensurable in square only.
I say that they also contain a rational rectangle. | ||
| Since A is to B as C is to D, therefore, alternately, A is to C as B is to D. | V.16 | |
| But A is to C as C is to B, therefore C is to B as B is to D. Therefore the rectangle C by D equals the square on B. But the square on B is rational, therefore the rectangle C by D is also rational.
Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. | ||
| Q.E.D. | ||
