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