A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.

Let *AB* be the side of the sum of two medial areas, and *CD* commensurable with *AB*.

It is to be proved that *CD* is also the side of the sum of two medial areas.

Since *AB* is the side of the sum of two medial areas, divide it into its straight lines at *E*, therefore *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 furthermore the sum of the squares on *AE* and *EB* incommensurable with the rectangle *AE* by *EB*.

Make the same construction as before.

We can then prove similarly that *CF* and *FD* are also incommensurable in square, the sum of the squares on *AE* and *EB* is commensurable with the sum of the squares on *CF* and *FD*, and the rectangle *AE* by *EB* with the rectangle *CF* by *FD*, so that the sum of the squares on *CF* and *FD* is also medial, the rectangle *CF* by *FD* is medial, and moreover the sum of the squares on *CF* and *FD* is incommensurable with the rectangle *CF* by *FD*.

Therefore *CD* is the side of the sum of two medial areas.

Therefore, *a straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.*

Q.E.D.