Since AF is incommensurable in length with FB, therefore the rectangle BA, AF is also incommensurable with the rectangle AB, BF. [X. 11]
But the rectangle BA, AF is equal to the square on AD, and the rectangle AB, BF to the square on DB;
therefore the square on AD is also incommensurable with the square on DB.
And, since the square on AB is medial, therefore the sum of the squares on AD, DB is also medial. [III. 31, I. 47]
And, since BC is double of DF, therefore the rectangle AB, BC is also double of the rectangle AB, FD.
But the rectangle AB, BC is rational;
therefore the rectangle AB, FD is also rational. [X. 6]
But the rectangle AB, FD is equal to the rectangle AD, DB; [Lemma]
so that the rectangle AD, DB is also rational.