If a straight line, which is incommensurable in square with the whole, and with the whole makes the sum of the squares on them medial, and twice the (rectangle contained) by them rational, is subtracted from a(nother) straight line then the remainder is an irrational (straight line). Let it be called that which makes with a rational (area) a medial whole. * For let the straight line $BC$, which is incommensurable in square with $AB$, and fulfils the (other) prescribed (conditions), have been subtracted from the straight line $AB$ [Prop. 10.34]. * I say that the remainder $AC$ is the aforementioned irrational (straight line).
Thus, that which makes with a rational (area) a medial whole is a straight line, whose length is expressible as
\[\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} -\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},\] for some positive rational number \(\rho\). See also [Prop. 10.40].
Proofs: 1
Proofs: 1 2 3 4
Propositions: 5 6