A major straight line is divided at one point only.

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

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

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

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

Therefore a major straight line is not divided at different points. Therefore it is only divided at one and the same point.

Therefore, *a major straight line is divided at one point only.*

Q.E.D.