Let the same construction be made as before shown.
Then, since AB is a second bimedial divided at C,
therefore AC, CB are medial straight lines commensurable in square only and containing a medial rectangle, [X. 38]
so that the sum of the squares on AC, CB is also medial. [X. 15 and 23 Por.]
And it is equal to DL; therefore DL is also medial.
And it is applied to the rational straight line DE;
therefore MD is also rational and incommensurable in length with DE. [X. 22]
For the same reason, MG is also rational and incommensurable in length with ML, that is, with DE;
therefore each of the straight lines DM, MG is rational and incommensurable in length with DE.
And, since AC is incommensurable in length with CB, and, as AC is to CB, so is the square on AC to the rectangle AC, CB,
therefore the square on AC is also incommensurable with the rectangle AC, CB. [X. 11]
Hence the sum of the squares on AC, CB is incommensurable with twice the rectangle AC, CB, [X. 12, 13]
that is, DL is incommensurable with MF, so that DM is also incommensurable with MG. [VI. 1, X. 11]
And they are rational; therefore DG is binomial. [X. 36]
It is to be proved that it is also a third binomial straight line.
Complete text
Book I