# Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely

### (Proposition 47 from Book 10 of Euclid's “Elements”)

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).

### Modern Formulation

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.

### Notes

This proposition corresponds to [Prop. 10.84], with plus signs instead of minus signs.

Proofs: 1

Propositions: 1

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### References

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

#### Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016