A first bimedial straight line is divided at one and the same point only.

Let *AB* be a first bimedial straight line divided at *C*, so that *AC* and *CB* are medial straight lines commensurable in square only and containing a rational rectangle.

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 also medial straight lines commensurable in square only and containing a rational rectangle.

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

Therefore a first bimedial straight line is not divided into its terms at different points. Therefore it is so divided at one point only.

Therefore, *a first bimedial straight line is divided at one and the same point only.*

Q.E.D.