The side of a rational plus a medial area is divided at one point only.

Let *AB* be the side of a rational plus a medial area divided at *C*, so that *AC* and *CB* are incommensurable in square and let the sum of the squares on *AC* and *CB* be medial, but twice the rectangle *AC* by *CB* rational.

I say that *AB* is not so divided at another point.

If possible, let it be divided at *D* also, so that *AD* and *DB* are also incommensurable in square and the sum of the squares on *AD* and *DB* is medial, but twice the rectangle *AD* by *DB* rational.

Since, then, that by which twice the rectangle *AC* by *CB* differs from twice the rectangle *AD* by *DB* is also that by which the sum of the squares on *AD* and *DB* differs from the sum of the squares on *AC* and *CB*, while twice the rectangle *AC* by *CB* exceeds twice the rectangle *AD* by *DB* by a rational area, therefore the sum of the squares on *AD* and *DB* also exceeds the sum of the squares on *AC* and *CB* by a rational area, though they are medial, which is impossible.

Therefore the side of a rational plus a medial area is not divided at different points, therefore it is divided at one point only.

Therefore, *the side of a rational plus a medial area is divided at one point only.*

Q.E.D.