# Definition:Side of Rational plus Medial Area

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## Definition

Let $a, b \in \R_{>0}$ be in the forms:

- $a = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} + k}$
- $b = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} - k}$

where:

- $\rho$ is a rational number
- $k$ is a rational number whose square root is irrational
- $\lambda$ is a rational number whose square root is irrational.

Then $a + b$ is the **side of the sum of (two) medial areas**.

In the words of Euclid:

*If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole straight line is irrational; and let it be called the***side of a rational plus a medial area***.*

(*The Elements*: Book $\text{X}$: Proposition $40$)