# Proposition 69

A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

Let AB be the side of a rational plus a medial area, and let CD be commensurable with AB.

It is to be proved that CD is also the side of a rational plus a medial area.

X.40

Divide AB into its straight lines at E. Then AE and EB are straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

Make the same construction as before.

We can then prove similarly that CF and FD are incommensurable in square, and the sum of the squares on AE and EB is commensurable with the sum of the squares on CF and FD, and the rectangle AE by EB with the rectangle CF by FD, so that the sum of the squares on CF and FD is also medial, and the rectangle CF by FD rational. Therefore CD is the side of a rational plus a medial area.

Therefore, a straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

Q.E.D.

## Guide

This proposition is not used in the rest of the Elements