If two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational are added together, then the whole straight line is irrational; let it be called the *side of a rational plus a medial area.*

Let two straight lines *AB* and *BC* incommensurable in square, and fulfilling the given conditions, be added together.

I say that *AC* is irrational.

Since the sum of the squares on *AB* and *BC* is medial, while twice the rectangle *AB* by *BC* is rational, therefore the sum of the squares on *AB* and *BC* is incommensurable with twice the rectangle *AB* by *BC*, so that the square on *AC* is also incommensurable with twice the rectangle *AB* by *BC*.

But twice the rectangle *AB* by *BC* is rational, therefore the square on *AC* is irrational. Therefore *AC* is irrational. Let it be called the *side of a rational plus a medial area.*

Q.E.D.