Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome

(Proposition 113 from Book 10 of Euclid's “Elements”)

The (square) on a rational (straight line), applied to an apotome, produces as breadth a binomial whose terms are commensurable with the terms of the apotome, and in the same ratio. Moreover, the created binomial has the same order as the apotome.

Let $A$ be a rational (straight line), and $BD$ an apotome.
And let the (rectangle contained) by $BD$ and $KH$ be equal to the (square) on $A$, such that the square on the rational (straight line) $A$, applied to the apotome $BD$, produces $KH$ as breadth.
I say that $KH$ is a binomial whose terms are commensurable with the terms of $BD$, and in the same ratio, and, moreover, that $KH$ has the same order as $BD$.