For, if possible, let DB also be so annexed;
therefore AD, DB are medial straight lines commensurable in square only and such that the rectangle AD, DB which they contain is rational. [X. 74]
Now, since the excess of the squares on AD, DB over twice the rectangle AD, DB is also the excess of the squares on AC, CB over twice the rectangle AC, CB,
for they exceed by the same, the square on AB, [II. 7]
therefore, alternately, the excess of the squares on AD, DB over the squares on AC, CB is also the excess of twice the rectangle AD, DB over twice the rectangle AC, CB.
But twice the rectangle AD, DB exceeds twice the rectangle AC, CB by a rational area, for both are rational.
Therefore the squares on AD, DB also exceed the squares on AC, CB by a rational area. which is impossible, for both are medial [X. 15 and 23, Por.],
and a medial area does not exceed a medial by a rational area. [X. 26]
Complete text
Book I