For let the same construction be made as before.
Then, since AB is the side of the sum of two medial areas, divided at C,
therefore AC, CB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and moreover the sum of the squares on them incommensurable with the rectangle contained by them, [X. 41]
so that, in accordance with what was before proved, each of the rectangles DL, MF is medial.
And they are applied to the rational straight line DE;
therefore each of the straight lines DM, MG is rational and incommensurable in length with DE. [X. 22]
And, since the sum of the squares on AC, CB is incommensurable with twice the rectangle AC, CB,
therefore DL is incommensurable with MF.
Therefore DM is also incommensurable with MG; [VI. 1, X. 11]
therefore DM, MG are rational straight lines commensurable in square only;
therefore DG is binomial. [X. 36]
I say next that it is also a sixth binomial straight line.