A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

Let *AB* be the side of a rational plus a medial area, and let *CD* be commensurable with *AB*.

It is to be proved that *CD* is also the side of a rational plus a medial area.

Divide *AB* into its straight lines at *E*. Then *AE* and *EB* are straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

Make the same construction as before.

We can then prove similarly that *CF* and *FD* are incommensurable in square, and 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, and the rectangle *CF* by *FD* rational. Therefore *CD* is the side of a rational plus a medial area.

Therefore, *a straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.*

Q.E.D.