The square root of (the sum of) two medial (areas) can be divided (into its component terms) at one point only. * Let $AB$ be [the square root of (the sum of) two medial (areas)] which has been divided at $C$, such that $AC$ and $CB$ are incommensurable in square, making the sum of the (squares) on $AC$ and $CB$ medial, and the (rectangle contained) by $AC$ and $CB$ medial, and, moreover, incommensurable with the sum of the (squares) on ($AC$ and $CB$) [Prop. 10.41]. * I say that $AB$ cannot be divided at another point fulfilling the prescribed (conditions).
In other words, \[\beta^{1/4}\sqrt{\frac{1+\alpha}{2\sqrt{1+\alpha^2}}}+\beta^{1/4}\sqrt{\frac{1-\alpha}{2\sqrt{1+\alpha^2}}}=\gamma^{1/4}\sqrt{\frac{1+\delta}{2\sqrt{1+\delta^2}}} + \gamma^{1/4}\sqrt{\frac{1-\delta}{2\sqrt{1+\delta^2}}}\] has only one solution: i.e., \[\delta=\alpha\quad\text{ and }\quad\gamma=\beta,\]
where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers.
This proposition corresponds to [Prop. 10.84], with plus signs instead of minus signs.
Proofs: 1
Propositions: 1
