For on AB let the square AD be described;
therefore AD is rational. [X. Def. 4]
And, since AB is commensurable in length with BC, while AB is equal to BD, therefore BD is commensurable in length with BC.
And, as BD is to BC, so is DA to AC. [VI. 1]
Therefore DA is commensurable with AC. [X. 11]
But DA is rational;
therefore AC is also rational. [X. Def. 4]