# Proposition 107

A straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole.

Let AB be a straight line which produces with a medial area a medial whole, and let CD be commensurable with AB.

I say that CD is also a straight line which produces with a medial area a medial whole.

Let BE be the annex to AB, and make the same construction.

X.78

Then AE and EB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them.

Now as was proved, AE and EB are commensurable with CF and FD, the sum of the squares on AE and EB with the sum of the squares on CF and FD, and the rectangle AE by EB with the rectangle CF by FD, therefore CF and FD are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them.

X.78

Therefore CD is a straight line which produces with a medial area a medial whole.

Therefore, a straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole.

Q.E.D.

## Guide

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