The rectangle contained by medial straight lines commensurable in length is medial.

Let the rectangle *AC* be contained by the medial straight lines *AB* and *BC* which are commensurable in length.

I say that *AC* is medial.

Describe the square *AD* on *AB*. Then *AD* is medial.

And, since *AB* is commensurable in length with *BC*, while *AB* equals *BD*, therefore *DB* is commensurable in length with *BC*, so that *DA* is commensurable with *AC*.

But *DA* is medial, therefore *AC* is also medial.

Therefore, *the rectangle contained by medial straight lines commensurable in length is medial.*

Q.E.D.

Medial lines were defined in Proposition X.21, but medial rectangles have not been defined. The meaning of the term medial rectangle must be interpreted by the way it is used in the proof of this proposition.

The square *AD*, which is the square on the medial line *AB*, is called a medial square, and since the rectangle *AC* is equal to the square *AD*, it is also proclaimed to be medial. Thus, a medial rectangle is one equal to the square on a medial line.

In this proposition, we have two medial numbers, *a* and *b*, such that *a*/*b* is a rational number. The conclusion is that *c* = √(*ab*) is a medial number. That is so since *c*^{2} = *ab* = *b*^{2}(*a*/*b*), which is an irrational number, while *c*^{4} = *a*^{2}*b*^{2} = *b*^{4}(*a*/*b*)^{2}, which is a rational number.

This proposition is not used in the rest of the *Elements*.