For, since the sum of the squares on AB, BC is medial, while twice the rectangle AB, BC is rational, therefore the sum of the squares on AB, BC is incommensurable with twice the rectangle AB, BC;
so that the square on AC is also incommensurable with twice the rectangle AB, BC. [X. 16 ]
But twice the rectangle AB, BC is rational;
therefore the square on AC is irrational.
Therefore AC is irrational. [X. Def. 4 ]
And let it be called the side of a rational plus a medial area.
Q. E. D.