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The rectangle contained by rational straight lines commensurable in length is rational.

For let the rectangle AC be contained by the rational straight lines AB, BC commensurable in length; I say that AC is rational.

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]

Therefore etc.

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