A straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole.

Let *AB* be a straight line which produces with a medial area a medial whole, and let *CD* be commensurable with *AB*.

I say that *CD* is also a straight line which produces with a medial area a medial whole.

Let *BE* be the annex to *AB*, and make the same construction.

Then *AE* and *EB* are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them.

Now as was proved, *AE* and *EB* are commensurable with *CF* and *FD*, the sum of the squares on *AE* and *EB* with the sum of the squares on *CF* and *FD*, and the rectangle *AE* by *EB* with the rectangle *CF* by *FD*, therefore *CF* and *FD* are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them.

Therefore *CD* is a straight line which produces with a medial area a medial whole.

Therefore, *a straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole.*

Q.E.D.