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 medial, and, moreover, the (sum of the) squares on them incommensurable with twice the (rectangle contained) by them, is subtracted from a(nother) straight line then the remainder is an irrational (straight line). Let it be called that which makes with a medial (area) a medial whole. * For let the straight line $BC$, which is incommensurable in square $AB$, and fulfils the (other) prescribed (conditions), have been subtracted from the (straight line) $AB$ [Prop. 10.35]. * I say that the remainder $AC$ is the irrational (straight line) called that which makes with a medial (area) a medial whole.
Thus, that which makes with a medial (area) a medial whole is a straight line, whose length is expressible as
\[\beta^{1/4}\left(\sqrt{\frac 12+\frac{\alpha}{2\sqrt{1+\alpha^2}}} -\sqrt{\frac 12-\frac{\alpha}{2\sqrt{1+\alpha^2}}}\right)\] for some positive rational numbers \(\alpha, \beta\). See also [Prop. 10.41].
Proofs: 1
Proofs: 1 2 3 4
Propositions: 5 6