For, since the sum of the squares on AB, BC is medial, while twice the rectangle AB, BC is rational, therefore the squares on AB, BC are incommensurable with twice the rectangle AB, BC;
therefore the remainder also, the square on AC, is incommensurable with twice the rectangle AB, BC. [II. 7, X. 16]
And twice the rectangle AB, BC is rational;
therefore the square on AC is irrational;
therefore AC is irrational.
And let it be called that which produces with a rational area a medial whole.
Q. E. D.