Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line

(Proposition 64 from Book 10 of Euclid's “Elements”)

The square on the square root of a rational plus a medial (area) applied to a rational (straight line) produces as breadth a fifth binomial (straight line).¹

Let $AB$ be the square root of a rational plus a medial (area) having been divided into its (component) straight lines at $C$, such that $AC$ is greater.
And let the rational (straight line) $DE$ be laid down.
And let the (parallelogram) $DF$, equal to the (square) on $AB$, have been applied to $DE$, producing $DG$ as breadth.
I say that $DG$ is a fifth binomial straight line.