A binomial (straight line) can be divided into its (component) terms at one point only.
In other words, \[\alpha + \sqrt{\beta} = \gamma + \sqrt{\delta}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad\delta=\beta,\] where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers. Likewise, \[\sqrt{\alpha} + \sqrt{\beta} =\sqrt{\gamma}+\sqrt{\delta}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad\delta=\beta,\] or, equivalently, \[\gamma=\beta\quad\text{ and }\quad\delta=\alpha.\]
This proposition corresponds to [Prop. 10.79], with plus signs instead of minus signs.
Proofs: 1